diff --git a/docs/reference-guides/core-blocks.md b/docs/reference-guides/core-blocks.md
index afc50e46ad2d59..c79c440c3a2168 100644
--- a/docs/reference-guides/core-blocks.md
+++ b/docs/reference-guides/core-blocks.md
@@ -311,7 +311,7 @@ Introduce new sections and organize content to help visitors (and search engines
 
 -	**Name:** core/heading
 -	**Category:** text
--	**Supports:** __unstablePasteTextInline, align (full, wide), anchor, className, color (background, gradients, link, text), spacing (margin, padding), typography (fontSize, lineHeight)
+-	**Supports:** align (full, wide), anchor, className, color (background, gradients, link, text), spacing (margin, padding), typography (fontSize, lineHeight)
 -	**Attributes:** content, level, placeholder, textAlign
 
 ## Home Link
diff --git a/packages/block-editor/src/components/rich-text/use-paste-handler.js b/packages/block-editor/src/components/rich-text/use-paste-handler.js
index 67c932aceddcc1..d86691ced70978 100644
--- a/packages/block-editor/src/components/rich-text/use-paste-handler.js
+++ b/packages/block-editor/src/components/rich-text/use-paste-handler.js
@@ -129,29 +129,6 @@ export function usePasteHandler( props ) {
 			}
 
 			const files = [ ...getFilesFromDataTransfer( clipboardData ) ];
-			const isInternal = clipboardData.getData( 'rich-text' ) === 'true';
-
-			// If the data comes from a rich text instance, we can directly use it
-			// without filtering the data. The filters are only meant for externally
-			// pasted content and remove inline styles.
-			if ( isInternal ) {
-				const pastedMultilineTag =
-					clipboardData.getData( 'rich-text-multi-line-tag' ) ||
-					undefined;
-				let pastedValue = create( {
-					html,
-					multilineTag: pastedMultilineTag,
-					multilineWrapperTags:
-						pastedMultilineTag === 'li'
-							? [ 'ul', 'ol' ]
-							: undefined,
-					preserveWhiteSpace,
-				} );
-				pastedValue = adjustLines( pastedValue, !! multilineTag );
-				addActiveFormats( pastedValue, value.activeFormats );
-				onChange( insert( value, pastedValue ) );
-				return;
-			}
 
 			if ( pastePlainText ) {
 				onChange( insert( value, create( { text: plainText } ) ) );
@@ -238,6 +215,10 @@ export function usePasteHandler( props ) {
 				mode,
 				tagName,
 				preserveWhiteSpace,
+				// If the data comes from a rich text instance, we can directly
+				// use it without filtering the data. The filters are only meant
+				// for externally pasted content and remove inline styles.
+				disableFilters: !! clipboardData.getData( 'rich-text' ),
 			} );
 
 			if ( typeof content === 'string' ) {
diff --git a/packages/block-library/src/heading/block.json b/packages/block-library/src/heading/block.json
index 80f1f0c47b5a8f..9c544e8f95c4f4 100644
--- a/packages/block-library/src/heading/block.json
+++ b/packages/block-library/src/heading/block.json
@@ -61,7 +61,6 @@
 				"textTransform": true
 			}
 		},
-		"__unstablePasteTextInline": true,
 		"__experimentalSlashInserter": true
 	},
 	"editorStyle": "wp-block-heading-editor",
diff --git a/packages/blocks/README.md b/packages/blocks/README.md
index 91cfec30c6a726..01547ca24ef683 100644
--- a/packages/blocks/README.md
+++ b/packages/blocks/README.md
@@ -458,6 +458,7 @@ _Parameters_
 -   _options.mode_ `[string]`: Handle content as blocks or inline content. _ 'AUTO': Decide based on the content passed. _ 'INLINE': Always handle as inline content, and return string. \* 'BLOCKS': Always handle as blocks, and return array of blocks.
 -   _options.tagName_ `[Array]`: The tag into which content will be inserted.
 -   _options.preserveWhiteSpace_ `[boolean]`: Whether or not to preserve consequent white space.
+-   _options.disableFilters_ `[boolean]`: Whether or not to filter non semantic content.
 
 _Returns_
 
diff --git a/packages/blocks/src/api/raw-handling/paste-handler.js b/packages/blocks/src/api/raw-handling/paste-handler.js
index c4ad40e0b1f509..f0acfac9fa7614 100644
--- a/packages/blocks/src/api/raw-handling/paste-handler.js
+++ b/packages/blocks/src/api/raw-handling/paste-handler.js
@@ -8,7 +8,6 @@ import { getPhrasingContentSchema, removeInvalidHTML } from '@wordpress/dom';
  */
 import { htmlToBlocks } from './html-to-blocks';
 import { hasBlockSupport } from '../registration';
-import { getBlockInnerHTML } from '../serializer';
 import parse from '../parser';
 import normaliseBlocks from './normalise-blocks';
 import specialCommentConverter from './special-comment-converter';
@@ -66,6 +65,40 @@ function filterInlineHTML( HTML, preserveWhiteSpace ) {
 	return HTML;
 }
 
+/**
+ * If we're allowed to return inline content, and there is only one inlineable
+ * block, and the original plain text content does not have any line breaks,
+ * then treat it as inline paste.
+ *
+ * @param {Object} options
+ * @param {Array}  options.blocks
+ * @param {string} options.plainText
+ * @param {string} options.mode
+ */
+function maybeConvertToInline( { blocks, plainText, mode } ) {
+	if (
+		mode === 'AUTO' &&
+		blocks.length === 1 &&
+		hasBlockSupport( blocks[ 0 ].name, '__unstablePasteTextInline', false )
+	) {
+		const trimRegex = /^[\n]+|[\n]+$/g;
+		// Don't catch line breaks at the start or end.
+		const trimmedPlainText = plainText.replace( trimRegex, '' );
+
+		if (
+			trimmedPlainText !== '' &&
+			trimmedPlainText.indexOf( '\n' ) === -1
+		) {
+			const target = blocks[ 0 ].innerBlocks.length
+				? blocks[ 0 ].innerBlocks[ 0 ]
+				: blocks[ 0 ];
+			return target.attributes.content;
+		}
+	}
+
+	return blocks;
+}
+
 /**
  * Converts an HTML string to known blocks. Strips everything else.
  *
@@ -79,6 +112,7 @@ function filterInlineHTML( HTML, preserveWhiteSpace ) {
  * @param {Array}   [options.tagName]            The tag into which content will be inserted.
  * @param {boolean} [options.preserveWhiteSpace] Whether or not to preserve consequent white space.
  *
+ * @param {boolean} [options.disableFilters]     Whether or not to filter non semantic content.
  * @return {Array|string} A list of blocks or a string, depending on `handlerMode`.
  */
 export function pasteHandler( {
@@ -87,6 +121,7 @@ export function pasteHandler( {
 	mode = 'AUTO',
 	tagName,
 	preserveWhiteSpace,
+	disableFilters,
 } ) {
 	// First of all, strip any meta tags.
 	HTML = HTML.replace( /<meta[^>]+>/g, '' );
@@ -121,6 +156,14 @@ export function pasteHandler( {
 		HTML = HTML.normalize();
 	}
 
+	if ( disableFilters ) {
+		return maybeConvertToInline( {
+			blocks: htmlToBlocks( normaliseBlocks( HTML ), pasteHandler ),
+			plainText,
+			mode,
+		} );
+	}
+
 	// Parse Markdown (and encoded HTML) if:
 	// * There is a plain text version.
 	// * There is no HTML version, or it has no formatting.
@@ -219,28 +262,5 @@ export function pasteHandler( {
 		.flat()
 		.filter( Boolean );
 
-	// If we're allowed to return inline content, and there is only one
-	// inlineable block, and the original plain text content does not have any
-	// line breaks, then treat it as inline paste.
-	if (
-		mode === 'AUTO' &&
-		blocks.length === 1 &&
-		hasBlockSupport( blocks[ 0 ].name, '__unstablePasteTextInline', false )
-	) {
-		const trimRegex = /^[\n]+|[\n]+$/g;
-		// Don't catch line breaks at the start or end.
-		const trimmedPlainText = plainText.replace( trimRegex, '' );
-
-		if (
-			trimmedPlainText !== '' &&
-			trimmedPlainText.indexOf( '\n' ) === -1
-		) {
-			return removeInvalidHTML(
-				getBlockInnerHTML( blocks[ 0 ] ),
-				phrasingContentSchema
-			).replace( trimRegex, '' );
-		}
-	}
-
-	return blocks;
+	return maybeConvertToInline( { blocks, plainText, mode } );
 }
diff --git a/packages/e2e-tests/specs/editor/various/__snapshots__/rich-text.test.js.snap b/packages/e2e-tests/specs/editor/various/__snapshots__/rich-text.test.js.snap
index 5d9c235e1c6a2e..7705ff11cbff9d 100644
--- a/packages/e2e-tests/specs/editor/various/__snapshots__/rich-text.test.js.snap
+++ b/packages/e2e-tests/specs/editor/various/__snapshots__/rich-text.test.js.snap
@@ -24,6 +24,16 @@ exports[`RichText should apply multiple formats when selection is collapsed 1`]
 <!-- /wp:paragraph -->"
 `;
 
+exports[`RichText should copy/paste heading 1`] = `
+"<!-- wp:heading -->
+<h2 class="wp-block-heading">Heading</h2>
+<!-- /wp:heading -->
+
+<!-- wp:heading -->
+<h2 class="wp-block-heading">Heading</h2>
+<!-- /wp:heading -->"
+`;
+
 exports[`RichText should handle Home and End keys 1`] = `
 "<!-- wp:paragraph -->
 <p>-<strong>12</strong>+</p>
@@ -129,7 +139,11 @@ exports[`RichText should paste paragraph contents into list 1`] = `
 
 <!-- wp:list -->
 <ul><!-- wp:list-item -->
-<li>1<br>2</li>
+<li>1</li>
+<!-- /wp:list-item -->
+
+<!-- wp:list-item -->
+<li>2</li>
 <!-- /wp:list-item --></ul>
 <!-- /wp:list -->"
 `;
diff --git a/packages/e2e-tests/specs/editor/various/rich-text.test.js b/packages/e2e-tests/specs/editor/various/rich-text.test.js
index 41b2c7253dfa2e..ff651e61d52ea9 100644
--- a/packages/e2e-tests/specs/editor/various/rich-text.test.js
+++ b/packages/e2e-tests/specs/editor/various/rich-text.test.js
@@ -556,4 +556,15 @@ describe( 'RichText', () => {
 		// Expect: <strong>1</strong>-<em>2</em>
 		expect( await getEditedPostContent() ).toMatchSnapshot();
 	} );
+
+	test( 'should copy/paste heading', async () => {
+		await insertBlock( 'Heading' );
+		await page.keyboard.type( 'Heading' );
+		await pressKeyWithModifier( 'primary', 'a' );
+		await pressKeyWithModifier( 'primary', 'c' );
+		await page.keyboard.press( 'ArrowRight' );
+		await page.keyboard.press( 'Enter' );
+		await pressKeyWithModifier( 'primary', 'v' );
+		expect( await getEditedPostContent() ).toMatchSnapshot();
+	} );
 } );
diff --git a/packages/rich-text/src/component/use-copy-handler.js b/packages/rich-text/src/component/use-copy-handler.js
index 45a7e1e0cfb8e2..b25e64428a1b4d 100644
--- a/packages/rich-text/src/component/use-copy-handler.js
+++ b/packages/rich-text/src/component/use-copy-handler.js
@@ -29,18 +29,21 @@ export function useCopyHandler( props ) {
 
 			const selectedRecord = slice( record.current );
 			const plainText = getTextContent( selectedRecord );
-			const html = toHTMLString( {
+			const tagName = element.tagName.toLowerCase();
+
+			let html = toHTMLString( {
 				value: selectedRecord,
 				multilineTag,
 				preserveWhiteSpace,
 			} );
+
+			if ( tagName && tagName !== 'span' && tagName !== 'div' ) {
+				html = `<${ tagName }>${ html }</${ tagName }>`;
+			}
+
 			event.clipboardData.setData( 'text/plain', plainText );
 			event.clipboardData.setData( 'text/html', html );
 			event.clipboardData.setData( 'rich-text', 'true' );
-			event.clipboardData.setData(
-				'rich-text-multi-line-tag',
-				multilineTag || ''
-			);
 			event.preventDefault();
 
 			if ( event.type === 'cut' ) {
diff --git a/test/integration/blocks-raw-handling.test.js b/test/integration/blocks-raw-handling.test.js
index 2a31d0b0ceaa28..733ae308c851ae 100644
--- a/test/integration/blocks-raw-handling.test.js
+++ b/test/integration/blocks-raw-handling.test.js
@@ -291,9 +291,11 @@ describe( 'Blocks raw handling', () => {
 			HTML: '<h1>FOO</h1>',
 			plainText: 'FOO\n',
 			mode: 'AUTO',
-		} );
+		} )
+			.map( getBlockContent )
+			.join( '' );
 
-		expect( filtered ).toBe( 'FOO' );
+		expect( filtered ).toBe( '<h1 class="wp-block-heading">FOO</h1>' );
 		expect( console ).toHaveLogged();
 	} );