The Turkish sports newspaper report that the Black Eagles will step up efforts to sign the United keeper following Cenk Gonen parting ways with the club.
Beşiktaş could however, face competition from fellow Süper Lig outfit Antalyaspor although the Black Eagles should be able to beat the newly promoted side to his signature.
Valdes is almost certain to leave United over the summer following comments made by Louis Van Gaal.
Van Gaal was speaking during a press-conference where United’s new transfer Bastian Schweinsteiger was unveiled when he stated: “He [Valdes] is not selected because he don’t follow our philosophy. There is no place for that one player.
“Last year he refused to play with the second team.”
“When you are not willing to follow the principles of the philosophy there is only one way – out.”
Besiktas finished last in third place and booked a place in the Europa League group stages.
The Black Eagles are currently managed by Senol Gunes who replaced Slaven Bilic over the summer. The former Croatia national team manager joined West Ham over the summer after resigning from his role as Besiktas head coach.