Before you proceed to add WP loginout link in WP Menu, I will recommend you for using WordPress Child Theme.  If you are new to wordpress and learning WordPress tweaking then WP child theme is best for you to avoid disaster.

Add WP loginout link in WP Menu

  1. Open wordpress child theme functions.php file
  2. Copy and paste this below codes in functions.php file
        Add LogIn/LogOut Link in Menu
        function add_login_logout_link($items, $args) { 
        $loginoutlink = wp_loginout('index.php', false); 
        $items .= '<li>'. $loginoutlink .'</li>'; 
        return $items; 
        add_filter('wp_nav_menu_items', 'add_login_logout_link', 10, 2);
  3. And save it.