The German-born Turk came through the youth system at Borussia Dortmund, and played over 100 times for Jurgen Klopp's side.
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He is expected to remain a fringe player this season for Los Blancos, and in recent pre-season fixtures, Sahin replaced Fabio Coentrao at left-back, instead of playing in his usual midfield role.
A number of sides have been courting the player, including Turkish champions Galatasaray, whose coach, Fatih Terim, reportedly has a long-standing admiration for the player. Terim has already signed Hamit Altintop from the Spanish champions this summer.
Besiktas are the only team to have enquired directly to Sahin, but he has rejected an offer from the Black Eagles, and this development gives weight to speculation that he wants to stay and fight for a place in the Madrid team.
Jose Mourinho's side continue their pre-season campaign against AC Milan in the United States on Wednesday.