Ticket #25777: 25777.2.diff

src/wp-admin/includes/update.php

@@ -209 +209 @@
         if ( ! isset( $cur->response ) || $cur->response != 'upgrade' )
                 return false;
 
+        // Make sure the site admin update notice won't be displayed too.
+        remove_action( 'admin_notices', 'site_admin_notice' );
+        remove_action( 'network_admin_notices', 'site_admin_notice' );
+
         if ( current_user_can('update_core') ) {
                 $msg = sprintf( __('<a href="http://codex.wordpress.org/Version_%1$s">WordPress %1$s</a> is available! <a href="%2$s">Please update now</a>.'), $cur->current, network_admin_url( 'update-core.php' ) );
         } else {