diff --git a/packages/components/CHANGELOG.md b/packages/components/CHANGELOG.md
index 1342b27cc2c081..b0f7f841c0a476 100644
--- a/packages/components/CHANGELOG.md
+++ b/packages/components/CHANGELOG.md
@@ -20,6 +20,7 @@
 -   `DropdownMenuV2`: do not collapse suffix width ([#57238](https://github.com/WordPress/gutenberg/pull/57238)).
 -   `DateTimePicker`: Adjustment of the dot position on DayButton and expansion of the button area. ([#55502](https://github.com/WordPress/gutenberg/pull/55502)).
 -   `Modal`: Improve application of body class names ([#55430](https://github.com/WordPress/gutenberg/pull/55430)).
+-   `Tooltip`: add `hideOnBlur` prop ([#57204](https://github.com/WordPress/gutenberg/pull/57204)).
 
 ### Experimental
 
diff --git a/packages/components/src/tooltip/README.md b/packages/components/src/tooltip/README.md
index 9b214e8fc6b00e..343b8726e90012 100644
--- a/packages/components/src/tooltip/README.md
+++ b/packages/components/src/tooltip/README.md
@@ -35,6 +35,15 @@ The amount of time in milliseconds to wait before showing the tooltip.
 -   Required: No
 -   Default: `700`
 
+#### `hideOnBlur`: `boolean`
+
+By default, the tooltip will close when interacting with other elements in the same `window`. This means that the tooltip will stay open when the whole `window` loses focus — for example, this happens when the whole browser loses keyboard focus, or even when moving focus to/from an iframe.
+
+This flag, when set to `true`, will force the tooltip to always hide when the anchor is blurred.
+
+-   Required: No
+-   Default: `false`
+
 #### `hideOnClick`: `boolean`
 
 Option to hide the tooltip when the anchor is clicked.
diff --git a/packages/components/src/tooltip/index.tsx b/packages/components/src/tooltip/index.tsx
index 80407def54cd45..90a303330634a4 100644
--- a/packages/components/src/tooltip/index.tsx
+++ b/packages/components/src/tooltip/index.tsx
@@ -28,6 +28,7 @@ function Tooltip( props: TooltipProps ) {
 		children,
 		delay = TOOLTIP_DELAY,
 		hideOnClick = true,
+		hideOnBlur = false,
 		placement,
 		position,
 		shortcut,
@@ -66,12 +67,14 @@ function Tooltip( props: TooltipProps ) {
 
 	const tooltipStore = Ariakit.useTooltipStore( {
 		placement: computedPlacement,
-		timeout: delay,
+		showTimeout: delay,
+		hideTimeout: 0,
 	} );
 
 	return (
 		<>
 			<Ariakit.TooltipAnchor
+				onBlur={ hideOnBlur ? tooltipStore.hide : undefined }
 				onClick={ hideOnClick ? tooltipStore.hide : undefined }
 				store={ tooltipStore }
 				render={ isOnlyChild ? children : undefined }
diff --git a/packages/components/src/tooltip/test/index.tsx b/packages/components/src/tooltip/test/index.tsx
index 4d58498e278d36..4b294ac2ddf60b 100644
--- a/packages/components/src/tooltip/test/index.tsx
+++ b/packages/components/src/tooltip/test/index.tsx
@@ -50,28 +50,39 @@ describe( 'Tooltip', () => {
 		).not.toBeInTheDocument();
 	} );
 
-	it( 'should render the tooltip when focusing on the tooltip anchor via tab', async () => {
+	it( 'should show the tooltip when the tooltip anchor receives focus, and hide it when the anchor loses focus when the hideOnBlur prop is `true`', async () => {
 		const user = userEvent.setup();
 
-		render( <Tooltip { ...props } /> );
+		render( <Tooltip { ...props } hideOnBlur /> );
+
+		const tooltipAnchor = screen.getByRole( 'button', { name: /Button/i } );
 
 		await user.tab();
 
-		expect(
-			screen.getByRole( 'button', { name: /Button/i } )
-		).toHaveFocus();
+		expect( tooltipAnchor ).toHaveFocus();
 
 		expect(
 			await screen.findByRole( 'tooltip', { name: /tooltip text/i } )
 		).toBeVisible();
 
 		await cleanupTooltip( user );
+
+		expect( tooltipAnchor ).not.toHaveFocus();
+
+		expect(
+			screen.queryByRole( 'tooltip', { name: /tooltip text/i } )
+		).not.toBeInTheDocument();
 	} );
 
-	it( 'should render the tooltip when the tooltip anchor is hovered', async () => {
+	it( 'should show the tooltip when the tooltip anchor is hovered, and hide it when the anchor is not hovered anymore', async () => {
 		const user = userEvent.setup();
 
-		render( <Tooltip { ...props } /> );
+		render(
+			<>
+				<Tooltip { ...props } />
+				<button>Hover me</button>
+			</>
+		);
 
 		await user.hover( screen.getByRole( 'button', { name: /Button/i } ) );
 
@@ -79,6 +90,12 @@ describe( 'Tooltip', () => {
 			await screen.findByRole( 'tooltip', { name: /tooltip text/i } )
 		).toBeVisible();
 
+		await user.hover( screen.getByRole( 'button', { name: /Hover me/i } ) );
+
+		expect(
+			screen.queryByRole( 'tooltip', { name: /tooltip text/i } )
+		).not.toBeInTheDocument();
+
 		await cleanupTooltip( user );
 	} );
 
@@ -263,7 +280,7 @@ describe( 'Tooltip', () => {
 		await cleanupTooltip( user );
 	} );
 
-	it( 'esc should close modal even when tooltip is visible', async () => {
+	it( 'should close a parent Modal even when tooltip is visible when pressing the Escape key', async () => {
 		const user = userEvent.setup();
 		const onRequestClose = jest.fn();
 		render(
diff --git a/packages/components/src/tooltip/types.ts b/packages/components/src/tooltip/types.ts
index 8708ae7005f5b3..b9c3ce59da34c7 100644
--- a/packages/components/src/tooltip/types.ts
+++ b/packages/components/src/tooltip/types.ts
@@ -22,6 +22,17 @@ export type TooltipProps = {
 	 * @default true
 	 */
 	hideOnClick?: boolean;
+	/**
+	 * By default, the tooltip will close when interacting with other elements
+	 * in the same `window`. This means that the tooltip will stay open when the
+	 * whole `window` loses focus — for example, this happens when the whole
+	 * browser loses keyboard focus, or even when moving focus to/from an iframe.
+	 * This flag, when set to `true`, will force the tooltip to always hide when
+	 * the anchor is blurred.
+	 *
+	 * @default false
+	 */
+	hideOnBlur?: boolean;
 	/**
 	 * The amount of time in milliseconds to wait before showing the tooltip.
 	 *