Index: src/wp-includes/embed.php
===================================================================
--- src/wp-includes/embed.php	(revision 36739)
+++ src/wp-includes/embed.php	(working copy)
@@ -341,15 +341,19 @@
  * Adds oEmbed discovery links in the website <head>.
  *
  * @since 4.4.0
+ * @since x.x.x Output was adjusted to only embed if the post type supports it.
  */
 function wp_oembed_add_discovery_links() {
+	$post_type = get_post_type();
 	$output = '';
 
-	if ( is_singular() ) {
-		$output .= '<link rel="alternate" type="application/json+oembed" href="' . esc_url( get_oembed_endpoint_url( get_permalink() ) ) . '" />' . "\n";
+	if ( is_singular() && $post_type ) {
+		if ( is_post_type_embeddable( $post_type ) ) {
+			$output .= '<link rel="alternate" type="application/json+oembed" href="' . esc_url( get_oembed_endpoint_url( get_permalink() ) ) . '" />' . "\n";
 
-		if ( class_exists( 'SimpleXMLElement' ) ) {
-			$output .= '<link rel="alternate" type="text/xml+oembed" href="' . esc_url( get_oembed_endpoint_url( get_permalink(), 'xml' ) ) . '" />' . "\n";
+			if ( class_exists( 'SimpleXMLElement' ) ) {
+				$output .= '<link rel="alternate" type="text/xml+oembed" href="' . esc_url( get_oembed_endpoint_url( get_permalink(), 'xml' ) ) . '" />' . "\n";
+			}
 		}
 	}
 
@@ -357,10 +361,12 @@
 	 * Filter the oEmbed discovery links HTML.
 	 *
 	 * @since 4.4.0
+	 * @since x.x.x The `$post_type` parameter was added.
 	 *
-	 * @param string $output HTML of the discovery links.
+	 * @param string $output    HTML of the discovery links.
+	 * @param string $post_type Post type slug.
 	 */
-	echo apply_filters( 'oembed_discovery_links', $output );
+	echo apply_filters( 'oembed_discovery_links', $output, $post_type );
 }
 
 /**
@@ -510,10 +516,11 @@
  * Retrieves the oEmbed response data for a given post.
  *
  * @since 4.4.0
+ * @since x.x.x Output was adjusted to only embed if the post type supports it.
  *
- * @param WP_Post|int $post  Post object or ID.
- * @param int         $width The requested width.
- * @return array|false Response data on success, false if post doesn't exist.
+ * @param WP_Post|int  $post  Post object or ID.
+ * @param int          $width The requested width.
+ * @return array|false Response data on success, false if post doesn't exist, or if post is not embeddable.
  */
 function get_oembed_response_data( $post, $width ) {
 	$post = get_post( $post );
@@ -526,6 +533,10 @@
 		return false;
 	}
 
+	if ( ! is_post_type_embeddable( get_post_type( $post ) ) ) {
+		return false;
+	}
+
 	/**
 	 * Filter the allowed minimum and maximum widths for the oEmbed response.
 	 *
Index: src/wp-includes/post.php
===================================================================
--- src/wp-includes/post.php	(revision 36739)
+++ src/wp-includes/post.php	(working copy)
@@ -890,6 +890,7 @@
  * @since 2.9.0
  * @since 3.0.0 The `show_ui` argument is now enforced on the new post screen.
  * @since 4.4.0 The `show_ui` argument is now enforced on the post type listing screen and post editing screen.
+ * @since x.x.x Introduced the `is_embeddable` argument.
  *
  * @global array      $wp_post_types List of post types.
  * @global WP_Rewrite $wp_rewrite    Used for default feeds.
@@ -962,6 +963,8 @@
  *     @type bool|string $has_archive          Whether there should be post type archives, or if a string, the
  *                                             archive slug to use. Will generate the proper rewrite rules if
  *                                             $rewrite is enabled. Default false.
+ *     @type bool        $is_embeddable        Whether this post type should be embeddable through oEmbed.
+ *                                             Default is value of `$public`.
  *     @type bool|array  $rewrite              {
  *         Triggers the handling of rewrites for this post type. To prevent rewrite, set to false.
  *         Defaults to true, using $post_type as slug. To specify rewrite rules, an array can be
@@ -1039,6 +1042,7 @@
 		'register_meta_box_cb' => null,
 		'taxonomies'           => array(),
 		'has_archive'          => false,
+		'is_embeddable'        => null,
 		'rewrite'              => true,
 		'query_var'            => true,
 		'can_export'           => true,
@@ -1163,6 +1167,11 @@
 		add_permastruct( $post_type, "{$args->rewrite['slug']}/%$post_type%", $permastruct_args );
 	}
 
+	// If not set, default to the setting for public.
+	if ( null === $args->is_embeddable ) {
+		$args->is_embeddable = $args->public;
+	}
+
 	// Register the post type meta box if a custom callback was specified.
 	if ( $args->register_meta_box_cb )
 		add_action( 'add_meta_boxes_' . $post_type, $args->register_meta_box_cb, 10, 1 );
@@ -1709,6 +1718,25 @@
 }
 
 /**
+ * Determines whether a post type is considered "embeddable".
+ *
+ * For all post types the 'is_embeddable' value will be evaluated.
+ *
+ * @param string $post_type Post type name.
+ * @return bool Whether the post type should be considered embeddable. False otherwise.
+ */
+function is_post_type_embeddable( $post_type ) {
+	if ( ! post_type_exists( $post_type ) ) {
+		return false;
+	}
+
+	if ( $post_type = get_post_type_object( $post_type ) ) {
+		return $post_type->is_embeddable;
+	}
+	return false;
+}
+
+/**
  * Retrieve list of latest posts or posts matching criteria.
  *
  * The defaults are as follows:
Index: tests/phpunit/tests/oembed/discovery.php
===================================================================
--- tests/phpunit/tests/oembed/discovery.php	(revision 36739)
+++ tests/phpunit/tests/oembed/discovery.php	(working copy)
@@ -71,4 +71,19 @@
 
 		$this->assertEquals( $expected, get_echo( 'wp_oembed_add_discovery_links' ) );
 	}
+
+	/**
+	 * @ticket 35567
+	 */
+	function test_wp_oembed_add_discovery_links_non_embeddable_post_type_output_should_be_empty() {
+		registeR_post_type( 'not_embeddable', array( 'is_embeddable' => false ) );
+
+		$post = $this->factory()->post->create_and_get( array(
+			'post_type' => 'not_embeddable'
+		) );
+
+		$this->go_to( get_permalink( $post->ID ) );
+
+		$this->assertEmpty( wp_oembed_add_discovery_links() );
+	}
 }
Index: tests/phpunit/tests/oembed/getResponseData.php
===================================================================
--- tests/phpunit/tests/oembed/getResponseData.php	(revision 36739)
+++ tests/phpunit/tests/oembed/getResponseData.php	(working copy)
@@ -167,4 +167,17 @@
 		$this->assertArrayHasKey( 'thumbnail_height', $data );
 		$this->assertTrue( 400 >= $data['thumbnail_width'] );
 	}
+
+	/**
+	 * @ticket 35567
+	 */
+	function test_get_oembed_response_data_non_embeddable_post_type_should_return_false() {
+		register_post_type( 'not_emebeddable', array( 'is_embeddable' => false ) );
+
+		$post = self::factory()->post->create_and_get( array(
+			'post_type' => 'not_embeddable'
+		) );
+
+		$this->assertFalse( get_oembed_response_data( $post, 100 ) );
+	}
 }
