Index: src/wp-includes/rest-api/endpoints/class-wp-rest-taxonomies-controller.php
===================================================================
--- src/wp-includes/rest-api/endpoints/class-wp-rest-taxonomies-controller.php	(revision 42227)
+++ src/wp-includes/rest-api/endpoints/class-wp-rest-taxonomies-controller.php	(working copy)
@@ -190,6 +190,17 @@
 		);
 
 		$context = ! empty( $request['context'] ) ? $request['context'] : 'view';
+
+		if ( 'edit' === $context ) {
+			$data['visibility'] = array(
+				'public'             => $taxonomy->public,
+				'publicly_queryable' => $taxonomy->publicly_queryable,
+				'show_admin_column'  => $taxonomy->show_admin_column,
+				'show_in_nav_menus'  => $taxonomy->show_in_nav_menus,
+				'show_in_quick_edit' => $taxonomy->show_in_quick_edit,
+				'show_ui'            => $taxonomy->show_ui,
+			);
+		}
 		$data = $this->add_additional_fields_to_object( $data, $request );
 		$data = $this->filter_response_by_context( $data, $context );
 
@@ -283,6 +294,12 @@
 					'context'      => array( 'view', 'edit' ),
 					'readonly'     => true,
 				),
+				'visibility'       => array(
+					'description'  => __( 'The visibility settings for the taxomonmy.' ),
+					'type'         => 'object',
+					'context'      => array( 'edit' ),
+					'readonly'     => true,
+				),
 				'rest_base'            => array(
 					'description'  => __( 'REST base route for the taxonomy.' ),
 					'type'         => 'string',
Index: tests/phpunit/tests/rest-api/rest-taxonomies-controller.php
===================================================================
--- tests/phpunit/tests/rest-api/rest-taxonomies-controller.php	(revision 42227)
+++ tests/phpunit/tests/rest-api/rest-taxonomies-controller.php	(working copy)
@@ -144,7 +144,7 @@
 		$response = $this->server->dispatch( $request );
 		$data = $response->get_data();
 		$properties = $data['schema']['properties'];
-		$this->assertEquals( 9, count( $properties ) );
+		$this->assertEquals( 10, count( $properties ) );
 		$this->assertArrayHasKey( 'capabilities', $properties );
 		$this->assertArrayHasKey( 'description', $properties );
 		$this->assertArrayHasKey( 'hierarchical', $properties );
@@ -153,6 +153,7 @@
 		$this->assertArrayHasKey( 'slug', $properties );
 		$this->assertArrayHasKey( 'show_cloud', $properties );
 		$this->assertArrayHasKey( 'types', $properties );
+		$this->assertArrayHasKey( 'visibility', $properties );
 		$this->assertArrayHasKey( 'rest_base', $properties );
 	}
 
@@ -186,10 +187,18 @@
 			$this->assertEquals( $tax_obj->cap, $data['capabilities'] );
 			$this->assertEquals( $tax_obj->labels, $data['labels'] );
 			$this->assertEquals( $tax_obj->show_tagcloud, $data['show_cloud'] );
+
+			$this->assertEquals( $tax_obj->public, $data['visibility']['public'] );
+			$this->assertEquals( $tax_obj->publicly_queryable, $data['visibility']['publicly_queryable'] );
+			$this->assertEquals( $tax_obj->show_admin_column, $data['visibility']['show_admin_column'] );
+			$this->assertEquals( $tax_obj->show_in_nav_menus, $data['visibility']['show_in_nav_menus'] );
+			$this->assertEquals( $tax_obj->show_in_quick_edit, $data['visibility']['show_in_quick_edit'] );
+			$this->assertEquals( $tax_obj->show_ui, $data['visibility']['show_ui'] );
 		} else {
 			$this->assertFalse( isset( $data['capabilities'] ) );
 			$this->assertFalse( isset( $data['labels'] ) );
 			$this->assertFalse( isset( $data['show_cloud'] ) );
+			$this->assertFalse( isset( $data['visibility'] ) );
 		}
 	}