The Turkish sports paper states that Valdes will undergo an extensive medical examination before putting pen-to-paper on a deal with the Black Eagles.
The Spain international and World Cup winner is expected to leave United over the summer after falling out with Van Gaal over the summer.
Valdes has fallen down the pecking order behind first-choice keeper Sergio Romero and David De Gae.
Beşiktaş have agreed terms with Valdes sources close to the club told Turkish-football.com.
The Black Eagles will compete in the Europa League Group A this summer against Sporting Lisbon, Lokomotiv Moscow and Skanderbeu.