diff --git a/packages/block-editor/src/components/off-canvas-editor/block-select-button.js b/packages/block-editor/src/components/off-canvas-editor/block-select-button.js
index 51404f6a39dbc5..07be2a992c2549 100644
--- a/packages/block-editor/src/components/off-canvas-editor/block-select-button.js
+++ b/packages/block-editor/src/components/off-canvas-editor/block-select-button.js
@@ -12,7 +12,7 @@ import {
 	__experimentalTruncate as Truncate,
 } from '@wordpress/components';
 import { forwardRef } from '@wordpress/element';
-import { Icon, lockSmall as lock } from '@wordpress/icons';
+import { Icon, lockSmall as lock, chevronRightSmall } from '@wordpress/icons';
 import { SPACE, ENTER } from '@wordpress/keycodes';
 import { sprintf, __ } from '@wordpress/i18n';
 
@@ -70,6 +70,10 @@ function ListViewBlockSelectButton(
 		  )
 		: __( 'Edit' );
 
+	const hasChevron = [ 'core/navigation-submenu', 'post', 'page' ].includes(
+		blockInformation.name
+	);
+
 	return (
 		<>
 			<Button
@@ -119,6 +123,13 @@ function ListViewBlockSelectButton(
 							<Icon icon={ lock } />
 						</span>
 					) }
+					{ hasChevron && (
+						<Icon
+							icon={ chevronRightSmall }
+							className="block-editor-list-view-block-select-button__drilldown-indicator"
+							size={ 24 }
+						/>
+					) }
 				</HStack>
 			</Button>
 		</>
diff --git a/packages/block-editor/src/components/use-block-display-information/index.js b/packages/block-editor/src/components/use-block-display-information/index.js
index 950cb9c905b24c..d910bbc1690bbe 100644
--- a/packages/block-editor/src/components/use-block-display-information/index.js
+++ b/packages/block-editor/src/components/use-block-display-information/index.js
@@ -57,12 +57,14 @@ export default function useBlockDisplayInformation( clientId ) {
 			const match = getActiveBlockVariation( blockName, attributes );
 			const isSynced =
 				isReusableBlock( blockType ) || isTemplatePart( blockType );
+
 			const blockTypeInfo = {
 				isSynced,
 				title: blockType.title,
 				icon: blockType.icon,
 				description: blockType.description,
 				anchor: attributes?.anchor,
+				name: blockType.name,
 			};
 			if ( ! match ) return blockTypeInfo;
 
@@ -72,6 +74,7 @@ export default function useBlockDisplayInformation( clientId ) {
 				icon: match.icon || blockType.icon,
 				description: match.description || blockType.description,
 				anchor: attributes?.anchor,
+				name: match.name || blockType.name,
 			};
 		},
 		[ clientId ]