The Spaniard joined United on an 18-month deal as cover for David de Gea in January but started just one first-team match last season before being omitted from this summer's pre-season tour.
Valdes was not given a squad number by Louis van Gaal this season and has been pictured in the directors' box at Old Trafford alongside want-away goalkeepers De Gea and Anders Lindegaard.
The Daily Telegraph report that he has agreed personal terms with Besiktas and will fly out to complete the deal ahead of the weekend. United are expected to release him from his contract to accelerate his transfer out of the club.
