Index: src/wp-includes/class-wp-embed.php
===================================================================
--- src/wp-includes/class-wp-embed.php	(revision 37605)
+++ src/wp-includes/class-wp-embed.php	(working copy)
@@ -336,6 +336,7 @@
 
 		// Find URLs that are on their own line.
 		$content = preg_replace_callback( '|^(\s*)(https?://[^\s"]+)(\s*)$|im', array( $this, 'autoembed_callback' ), $content );
+		$content = preg_replace_callback( '|(<p[^>]*>\s*)(https?://[^\s<]+)(\s*<\/p>)|i', array( $this, 'autoembed_callback' ), $content );
 
 		// Put the line breaks back.
 		return str_replace( '<!-- wp-line-break -->', "\n", $content );
Index: tests/phpunit/tests/media.php
===================================================================
--- tests/phpunit/tests/media.php	(revision 37605)
+++ tests/phpunit/tests/media.php	(working copy)
@@ -158,6 +158,69 @@
 		$this->assertEquals( $content, $result );
 	}
 
+	function data_autoembed() {
+		return array(
+
+			// Should embed
+			array(
+'https://w.org',
+'[embed]'
+			),
+			array(
+'test
+ https://w.org
+test',
+'test
+ [embed]
+test'
+			),
+			array(
+'<p class="test">https://w.org</p>',
+'<p class="test">[embed]</p>'
+			),
+			array(
+'<p> https://w.org </p>',
+'<p> [embed] </p>'
+			),
+			array(
+'<p>test
+https://w.org
+test</p>',
+'<p>test
+[embed]
+test</p>'
+			),
+			array(
+'<p>https://w.org
+</p>',
+'<p>[embed]
+</p>'
+			),
+
+			// Should NOT embed
+			array(
+'test https://w.org</p>'
+			),
+			array(
+'<span>https://w.org</a>'
+			),
+			array(
+'<a href=" https://w.org ">
+https://w.org
+</a>'
+			)
+		);
+	}
+
+	/**
+	 * @dataProvider data_autoembed
+	 */
+	function test_autoembed( $content, $result = null ) {
+		$wp_embed = new Test_Autoembed;
+
+		$this->assertEquals( $wp_embed->autoembed( $content ), $result ? $result : $content );
+	}
+
 	function test_wp_prepare_attachment_for_js() {
 		// Attachment without media
 		$id = wp_insert_attachment(array(
@@ -1612,3 +1675,12 @@
 		$this->assertSame( $expected, get_image_send_to_editor( $id, $caption, $title, $align, $url, $rel, $size, $alt ) );
 	}
 }
+
+/**
+ * Helper class for `test_autoembed`.
+ */
+class Test_Autoembed extends WP_Embed {
+	public function shortcode( $attr, $url = '' ) {
+		return '[embed]';
+	}
+}