diff --git a/wp-admin/includes/theme.php b/wp-admin/includes/theme.php
index 5738893a9e..2fd6efe513 100644
--- a/wp-admin/includes/theme.php
+++ b/wp-admin/includes/theme.php
@@ -684,27 +684,34 @@ function wp_prepare_themes_for_js( $themes = null ) {
 			);
 		}
 
+		$update_requires_wp  = isset( $updates[ $slug ]['requires'] ) ? $updates[ $slug ]['requires'] : null;
+		$update_requires_php = isset( $updates[ $slug ]['requires_php'] ) ? $updates[ $slug ]['requires_php'] : null;
+
 		$auto_update        = in_array( $slug, $auto_updates, true );
 		$auto_update_action = $auto_update ? 'disable-auto-update' : 'enable-auto-update';
 
 		$prepared_themes[ $slug ] = array(
-			'id'            => $slug,
-			'name'          => $theme->display( 'Name' ),
-			'screenshot'    => array( $theme->get_screenshot() ), // @todo Multiple screenshots.
-			'description'   => $theme->display( 'Description' ),
-			'author'        => $theme->display( 'Author', false, true ),
-			'authorAndUri'  => $theme->display( 'Author' ),
-			'tags'          => $theme->display( 'Tags' ),
-			'version'       => $theme->get( 'Version' ),
-			'compatibleWP'  => is_wp_version_compatible( $theme->get( 'RequiresWP' ) ),
-			'compatiblePHP' => is_php_version_compatible( $theme->get( 'RequiresPHP' ) ),
-			'parent'        => $parent,
-			'active'        => $slug === $current_theme,
-			'hasUpdate'     => isset( $updates[ $slug ] ),
-			'hasPackage'    => isset( $updates[ $slug ] ) && ! empty( $updates[ $slug ]['package'] ),
-			'update'        => get_theme_update_available( $theme ),
-			'autoupdate'    => $auto_update,
-			'actions'       => array(
+			'id'             => $slug,
+			'name'           => $theme->display( 'Name' ),
+			'screenshot'     => array( $theme->get_screenshot() ), // @todo Multiple screenshots.
+			'description'    => $theme->display( 'Description' ),
+			'author'         => $theme->display( 'Author', false, true ),
+			'authorAndUri'   => $theme->display( 'Author' ),
+			'tags'           => $theme->display( 'Tags' ),
+			'version'        => $theme->get( 'Version' ),
+			'compatibleWP'   => is_wp_version_compatible( $theme->get( 'RequiresWP' ) ),
+			'compatiblePHP'  => is_php_version_compatible( $theme->get( 'RequiresPHP' ) ),
+			'updateResponse' => array(
+				'compatibleWP'  => is_wp_version_compatible( $update_requires_wp ),
+				'compatiblePHP' => is_php_version_compatible( $update_requires_php ),
+			),
+			'parent'         => $parent,
+			'active'         => $slug === $current_theme,
+			'hasUpdate'      => isset( $updates[ $slug ] ),
+			'hasPackage'     => isset( $updates[ $slug ] ) && ! empty( $updates[ $slug ]['package'] ),
+			'update'         => get_theme_update_available( $theme ),
+			'autoupdate'     => $auto_update,
+			'actions'        => array(
 				'activate'   => current_user_can( 'switch_themes' ) ? wp_nonce_url( admin_url( 'themes.php?action=activate&stylesheet=' . $encoded_slug ), 'switch-theme_' . $slug ) : null,
 				'customize'  => $customize_action,
 				'delete'     => current_user_can( 'delete_themes' ) ? wp_nonce_url( admin_url( 'themes.php?action=delete&stylesheet=' . $encoded_slug ), 'delete-theme_' . $slug ) : null,
@@ -793,10 +800,68 @@ function customize_themes_print_templates() {
 					<# } #>
 
 					<# if ( data.hasUpdate ) { #>
-						<div class="notice notice-warning notice-alt notice-large" data-slug="{{ data.id }}">
-							<h3 class="notice-title"><?php _e( 'Update Available' ); ?></h3>
-							{{{ data.update }}}
-						</div>
+						<# if ( data.updateResponse.compatibleWP && data.updateResponse.compatiblePHP ) { #>
+							<div class="notice notice-warning notice-alt notice-large" data-slug="{{ data.id }}">
+								<h3 class="notice-title"><?php _e( 'Update Available' ); ?></h3>
+								{{{ data.update }}}
+							</div>
+						<# } else { #>
+							<div class="notice notice-error notice-alt notice-large" data-slug="{{ data.id }}">
+								<h3 class="notice-title"><?php _e( 'Update Incompatible' ); ?></h3>
+								<p>
+									<# if ( ! data.updateResponse.compatibleWP && ! data.updateResponse.compatiblePHP ) { #>
+										<?php
+										_e( 'There is a new version available, but it doesn&#8217;t work with your versions of WordPress and PHP.' );
+										if ( current_user_can( 'update_core' ) && current_user_can( 'update_php' ) ) {
+											printf(
+												/* translators: 1: URL to WordPress Updates screen, 2: URL to Update PHP page. */
+												' ' . __( '<a href="%1$s">Please update WordPress</a>, and then <a href="%2$s">learn more about updating PHP</a>.' ),
+												self_admin_url( 'update-core.php' ),
+												esc_url( wp_get_update_php_url() )
+											);
+											wp_update_php_annotation( '</p><p><em>', '</em>' );
+										} elseif ( current_user_can( 'update_core' ) ) {
+											printf(
+												/* translators: %s: URL to WordPress Updates screen. */
+												' ' . __( '<a href="%s">Please update WordPress</a>.' ),
+												self_admin_url( 'update-core.php' )
+											);
+										} elseif ( current_user_can( 'update_php' ) ) {
+											printf(
+												/* translators: %s: URL to Update PHP page. */
+												' ' . __( '<a href="%s">Learn more about updating PHP</a>.' ),
+												esc_url( wp_get_update_php_url() )
+											);
+											wp_update_php_annotation( '</p><p><em>', '</em>' );
+										}
+										?>
+									<# } else if ( ! data.updateResponse.compatibleWP ) { #>
+										<?php
+										_e( 'There is a new version available, but it doesn&#8217;t work with your version of WordPress.' );
+										if ( current_user_can( 'update_core' ) ) {
+											printf(
+												/* translators: %s: URL to WordPress Updates screen. */
+												' ' . __( '<a href="%s">Please update WordPress</a>.' ),
+												self_admin_url( 'update-core.php' )
+											);
+										}
+										?>
+									<# } else if ( ! data.updateResponse.compatiblePHP ) { #>
+										<?php
+										_e( 'There is a new version available, but it doesn&#8217;t work with your version of PHP.' );
+										if ( current_user_can( 'update_php' ) ) {
+											printf(
+												/* translators: %s: URL to Update PHP page. */
+												' ' . __( '<a href="%s">Learn more about updating PHP</a>.' ),
+												esc_url( wp_get_update_php_url() )
+											);
+											wp_update_php_annotation( '</p><p><em>', '</em>' );
+										}
+										?>
+									<# } #>
+								</p>
+							</div>
+						<# } #>
 					<# } #>
 
 					<# if ( data.parent ) { #>
diff --git a/wp-admin/themes.php b/wp-admin/themes.php
index 03d917a070..280ba6be1e 100644
--- a/wp-admin/themes.php
+++ b/wp-admin/themes.php
@@ -389,13 +389,64 @@ foreach ( $themes as $theme ) :
 	<?php } ?>
 
 	<?php if ( $theme['hasUpdate'] ) : ?>
-		<div class="update-message notice inline notice-warning notice-alt">
-		<?php if ( $theme['hasPackage'] ) : ?>
-			<p><?php _e( 'New version available. <button class="button-link" type="button">Update now</button>' ); ?></p>
+		<?php if ( $theme['updateResponse']['compatibleWP'] && $theme['updateResponse']['compatiblePHP'] ) : ?>
+			<div class="update-message notice inline notice-warning notice-alt"><p>
+				<?php if ( $theme['hasPackage'] ) : ?>
+					<?php _e( 'New version available. <button class="button-link" type="button">Update now</button>' ); ?>
+				<?php else : ?>
+					<?php _e( 'New version available.' ); ?>
+				<?php endif; ?>
+			</p></div>
 		<?php else : ?>
-			<p><?php _e( 'New version available.' ); ?></p>
+			<div class="notice inline notice-error notice-alt"><p>
+				<?php
+				if ( $theme['updateResponse']['compatibleWP'] && $theme['updateResponse']['compatiblePHP'] ) {
+					_e( 'There is a new version available, but it doesn&#8217;t work with your versions of WordPress and PHP.' );
+					if ( current_user_can( 'update_core' ) && current_user_can( 'update_php' ) ) {
+						printf(
+							/* translators: 1: URL to WordPress Updates screen, 2: URL to Update PHP page. */
+							' ' . __( '<a href="%1$s">Please update WordPress</a>, and then <a href="%2$s">learn more about updating PHP</a>.' ),
+							self_admin_url( 'update-core.php' ),
+							esc_url( wp_get_update_php_url() )
+						);
+						wp_update_php_annotation( '</p><p><em>', '</em>' );
+					} elseif ( current_user_can( 'update_core' ) ) {
+						printf(
+							/* translators: %s: URL to WordPress Updates screen. */
+							' ' . __( '<a href="%s">Please update WordPress</a>.' ),
+							self_admin_url( 'update-core.php' )
+						);
+					} elseif ( current_user_can( 'update_php' ) ) {
+						printf(
+							/* translators: %s: URL to Update PHP page. */
+							' ' . __( '<a href="%s">Learn more about updating PHP</a>.' ),
+							esc_url( wp_get_update_php_url() )
+						);
+						wp_update_php_annotation( '</p><p><em>', '</em>' );
+					}
+				} elseif ( ! $theme['updateResponse']['compatibleWP'] ) {
+					_e( 'There is a new version available, but it doesn&#8217;t work with your version of WordPress.' );
+					if ( current_user_can( 'update_core' ) ) {
+						printf(
+							/* translators: %s: URL to WordPress Updates screen. */
+							' ' . __( '<a href="%s">Please update WordPress</a>.' ),
+							self_admin_url( 'update-core.php' )
+						);
+					}
+				} elseif ( ! $theme['updateResponse']['compatiblePHP'] ) {
+					_e( 'There is a new version available, but it doesn&#8217;t work with your version of PHP.' );
+					if ( current_user_can( 'update_php' ) ) {
+						printf(
+							/* translators: %s: URL to Update PHP page. */
+							' ' . __( '<a href="%s">Learn more about updating PHP</a>.' ),
+							esc_url( wp_get_update_php_url() )
+						);
+						wp_update_php_annotation( '</p><p><em>', '</em>' );
+					}
+				}
+				?>
+			</p></div>
 		<?php endif; ?>
-		</div>
 	<?php endif; ?>
 
 	<?php
@@ -659,10 +710,67 @@ function wp_theme_auto_update_setting_template() {
 	<# } #>
 
 	<# if ( data.hasUpdate ) { #>
-		<# if ( data.hasPackage ) { #>
-			<div class="update-message notice inline notice-warning notice-alt"><p><?php _e( 'New version available. <button class="button-link" type="button">Update now</button>' ); ?></p></div>
+		<# if ( data.updateResponse.compatibleWP && data.updateResponse.compatiblePHP ) { #>
+			<div class="update-message notice inline notice-warning notice-alt"><p>
+				<# if ( data.hasPackage ) { #>
+					<?php _e( 'New version available. <button class="button-link" type="button">Update now</button>' ); ?>
+				<# } else { #>
+					<?php _e( 'New version available.' ); ?>
+				<# } #>
+			</p></div>
 		<# } else { #>
-			<div class="update-message notice inline notice-warning notice-alt"><p><?php _e( 'New version available.' ); ?></p></div>
+			<div class="notice inline notice-error notice-alt"><p>
+				<# if ( ! data.updateResponse.compatibleWP && ! data.updateResponse.compatiblePHP ) { #>
+					<?php
+					_e( 'There is a new version available, but it doesn&#8217;t work with your versions of WordPress and PHP.' );
+					if ( current_user_can( 'update_core' ) && current_user_can( 'update_php' ) ) {
+						printf(
+							/* translators: 1: URL to WordPress Updates screen, 2: URL to Update PHP page. */
+							' ' . __( '<a href="%1$s">Please update WordPress</a>, and then <a href="%2$s">learn more about updating PHP</a>.' ),
+							self_admin_url( 'update-core.php' ),
+							esc_url( wp_get_update_php_url() )
+						);
+						wp_update_php_annotation( '</p><p><em>', '</em>' );
+					} elseif ( current_user_can( 'update_core' ) ) {
+						printf(
+							/* translators: %s: URL to WordPress Updates screen. */
+							' ' . __( '<a href="%s">Please update WordPress</a>.' ),
+							self_admin_url( 'update-core.php' )
+						);
+					} elseif ( current_user_can( 'update_php' ) ) {
+						printf(
+							/* translators: %s: URL to Update PHP page. */
+							' ' . __( '<a href="%s">Learn more about updating PHP</a>.' ),
+							esc_url( wp_get_update_php_url() )
+						);
+						wp_update_php_annotation( '</p><p><em>', '</em>' );
+					}
+					?>
+				<# } else if ( ! data.updateResponse.compatibleWP ) { #>
+					<?php
+					_e( 'There is a new version available, but it doesn&#8217;t work with your version of WordPress.' );
+					if ( current_user_can( 'update_core' ) ) {
+						printf(
+							/* translators: %s: URL to WordPress Updates screen. */
+							' ' . __( '<a href="%s">Please update WordPress</a>.' ),
+							self_admin_url( 'update-core.php' )
+						);
+					}
+					?>
+				<# } else if ( ! data.updateResponse.compatiblePHP ) { #>
+					<?php
+					_e( 'There is a new version available, but it doesn&#8217;t work with your version of PHP.' );
+					if ( current_user_can( 'update_php' ) ) {
+						printf(
+							/* translators: %s: URL to Update PHP page. */
+							' ' . __( '<a href="%s">Learn more about updating PHP</a>.' ),
+							esc_url( wp_get_update_php_url() )
+						);
+						wp_update_php_annotation( '</p><p><em>', '</em>' );
+					}
+					?>
+				<# } #>
+			</p></div>
 		<# } #>
 	<# } #>
 
@@ -853,15 +961,73 @@ function wp_theme_auto_update_setting_template() {
 					</p></div>
 				<# } #>
 
-				<# if ( data.actions.autoupdate ) { #>
-					<?php echo wp_theme_auto_update_setting_template(); ?>
+				<# if ( data.hasUpdate ) { #>
+					<# if ( data.updateResponse.compatibleWP && data.updateResponse.compatiblePHP ) { #>
+						<div class="notice notice-warning notice-alt notice-large">
+							<h3 class="notice-title"><?php _e( 'Update Available' ); ?></h3>
+							{{{ data.update }}}
+						</div>
+					<# } else { #>
+						<div class="notice notice-error notice-alt notice-large">
+							<h3 class="notice-title"><?php _e( 'Update Incompatible' ); ?></h3>
+							<p>
+								<# if ( ! data.updateResponse.compatibleWP && ! data.updateResponse.compatiblePHP ) { #>
+									<?php
+									_e( 'There is a new version available, but it doesn&#8217;t work with your versions of WordPress and PHP.' );
+									if ( current_user_can( 'update_core' ) && current_user_can( 'update_php' ) ) {
+										printf(
+											/* translators: 1: URL to WordPress Updates screen, 2: URL to Update PHP page. */
+											' ' . __( '<a href="%1$s">Please update WordPress</a>, and then <a href="%2$s">learn more about updating PHP</a>.' ),
+											self_admin_url( 'update-core.php' ),
+											esc_url( wp_get_update_php_url() )
+										);
+										wp_update_php_annotation( '</p><p><em>', '</em>' );
+									} elseif ( current_user_can( 'update_core' ) ) {
+										printf(
+											/* translators: %s: URL to WordPress Updates screen. */
+											' ' . __( '<a href="%s">Please update WordPress</a>.' ),
+											self_admin_url( 'update-core.php' )
+										);
+									} elseif ( current_user_can( 'update_php' ) ) {
+										printf(
+											/* translators: %s: URL to Update PHP page. */
+											' ' . __( '<a href="%s">Learn more about updating PHP</a>.' ),
+											esc_url( wp_get_update_php_url() )
+										);
+										wp_update_php_annotation( '</p><p><em>', '</em>' );
+									}
+									?>
+								<# } else if ( ! data.updateResponse.compatibleWP ) { #>
+									<?php
+									_e( 'There is a new version available, but it doesn&#8217;t work with your version of WordPress.' );
+									if ( current_user_can( 'update_core' ) ) {
+										printf(
+											/* translators: %s: URL to WordPress Updates screen. */
+											' ' . __( '<a href="%s">Please update WordPress</a>.' ),
+											self_admin_url( 'update-core.php' )
+										);
+									}
+									?>
+								<# } else if ( ! data.updateResponse.compatiblePHP ) { #>
+									<?php
+									_e( 'There is a new version available, but it doesn&#8217;t work with your version of PHP.' );
+									if ( current_user_can( 'update_php' ) ) {
+										printf(
+											/* translators: %s: URL to Update PHP page. */
+											' ' . __( '<a href="%s">Learn more about updating PHP</a>.' ),
+											esc_url( wp_get_update_php_url() )
+										);
+										wp_update_php_annotation( '</p><p><em>', '</em>' );
+									}
+									?>
+								<# } #>
+							</p>
+						</div>
+					<# } #>
 				<# } #>
 
-				<# if ( data.hasUpdate ) { #>
-				<div class="notice notice-warning notice-alt notice-large">
-					<h3 class="notice-title"><?php _e( 'Update Available' ); ?></h3>
-					{{{ data.update }}}
-				</div>
+				<# if ( data.actions.autoupdate ) { #>
+					<?php echo wp_theme_auto_update_setting_template(); ?>
 				<# } #>
 
 				<p class="theme-description">{{{ data.description }}}</p>
diff --git a/wp-includes/customize/class-wp-customize-theme-control.php b/wp-includes/customize/class-wp-customize-theme-control.php
index 64dc33a3e4..03d9649ff8 100644
--- a/wp-includes/customize/class-wp-customize-theme-control.php
+++ b/wp-includes/customize/class-wp-customize-theme-control.php
@@ -90,21 +90,78 @@ class WP_Customize_Theme_Control extends WP_Customize_Control {
 			</div>
 
 			<# if ( 'installed' === data.theme.type && data.theme.hasUpdate ) { #>
-				<div class="update-message notice inline notice-warning notice-alt" data-slug="{{ data.theme.id }}">
-					<p>
-						<?php
-						if ( is_multisite() ) {
-							_e( 'New version available.' );
-						} else {
-							printf(
-								/* translators: %s: "Update now" button. */
-								__( 'New version available. %s' ),
-								'<button class="button-link update-theme" type="button">' . __( 'Update now' ) . '</button>'
-							);
-						}
-						?>
-					</p>
-				</div>
+				<# if ( data.theme.updateResponse.compatibleWP && data.theme.updateResponse.compatiblePHP ) { #>
+					<div class="update-message notice inline notice-warning notice-alt" data-slug="{{ data.theme.id }}">
+						<p>
+							<?php
+							if ( is_multisite() ) {
+								_e( 'New version available.' );
+							} else {
+								printf(
+									/* translators: %s: "Update now" button. */
+									__( 'New version available. %s' ),
+									'<button class="button-link update-theme" type="button">' . __( 'Update now' ) . '</button>'
+								);
+							}
+							?>
+						</p>
+					</div>
+				<# } else { #>
+					<div class="update-message notice inline notice-warning notice-alt" data-slug="{{ data.theme.id }}">
+						<p>
+							<# if ( ! data.theme.updateResponse.compatibleWP && ! data.theme.updateResponse.compatiblePHP ) { #>
+								<?php
+								_e( 'There is a new version available, but it doesn&#8217;t work with your versions of WordPress and PHP.' );
+								if ( current_user_can( 'update_core' ) && current_user_can( 'update_php' ) ) {
+									printf(
+										/* translators: 1: URL to WordPress Updates screen, 2: URL to Update PHP page. */
+										' ' . __( '<a href="%1$s">Please update WordPress</a>, and then <a href="%2$s">learn more about updating PHP</a>.' ),
+										self_admin_url( 'update-core.php' ),
+										esc_url( wp_get_update_php_url() )
+									);
+									wp_update_php_annotation( '</p><p><em>', '</em>' );
+								} elseif ( current_user_can( 'update_core' ) ) {
+									printf(
+										/* translators: %s: URL to WordPress Updates screen. */
+										' ' . __( '<a href="%s">Please update WordPress</a>.' ),
+										self_admin_url( 'update-core.php' )
+									);
+								} elseif ( current_user_can( 'update_php' ) ) {
+									printf(
+										/* translators: %s: URL to Update PHP page. */
+										' ' . __( '<a href="%s">Learn more about updating PHP</a>.' ),
+										esc_url( wp_get_update_php_url() )
+									);
+									wp_update_php_annotation( '</p><p><em>', '</em>' );
+								}
+								?>
+							<# } else if ( ! data.theme.updateResponse.compatibleWP ) { #>
+								<?php
+								_e( 'There is a new version available, but it doesn&#8217;t work with your version of WordPress.' );
+								if ( current_user_can( 'update_core' ) ) {
+									printf(
+										/* translators: %s: URL to WordPress Updates screen. */
+										' ' . __( '<a href="%s">Please update WordPress</a>.' ),
+										self_admin_url( 'update-core.php' )
+									);
+								}
+								?>
+							<# } else if ( ! data.theme.updateResponse.compatiblePHP ) { #>
+								<?php
+								_e( 'There is a new version available, but it doesn&#8217;t work with your version of PHP.' );
+								if ( current_user_can( 'update_php' ) ) {
+									printf(
+										/* translators: %s: URL to Update PHP page. */
+										' ' . __( '<a href="%s">Learn more about updating PHP</a>.' ),
+										esc_url( wp_get_update_php_url() )
+									);
+									wp_update_php_annotation( '</p><p><em>', '</em>' );
+								}
+								?>
+							<# } #>
+						</p>
+					</div>
+				<# } #>
 			<# } #>
 
 			<# if ( ! data.theme.compatibleWP || ! data.theme.compatiblePHP ) { #>
diff --git a/wp-includes/version.php b/wp-includes/version.php
index 117519d114..11e9e1f0e5 100644
--- a/wp-includes/version.php
+++ b/wp-includes/version.php
@@ -13,7 +13,7 @@
  *
  * @global string $wp_version
  */
-$wp_version = '5.5-beta4-48651';
+$wp_version = '5.5-beta4-48652';
 
 /**
  * Holds the WordPress DB revision, increments when changes are made to the WordPress DB schema.