The 24-year-old Dutchman only signed for the reigning Premier League champions last summer from Vitesse Arnhem, but has struggled to establish himself in the Red Devils first-team.
Patrice Evra remains Manchester United's first-choice left-back, and with new manager David Moyes credited with an interest in signing Everton star Leighton Baines, Buttner's future looks increasingly bleak at Old Trafford.
If United are successful in their attempt to recruit the England international defender, then Besiktas will press ahead with a move for Buttner, in the hope that the offer of regular first-team football will be enough to convince him to swap England for Turkey.
After making just five Premier League appearances last season, the full-back will be determined not to see his progress halted by another spell on the sidelines, and so might be open to the possibility of a move before the transfer window closes on September 1.
Moyes was responsible for bringing Baines to Goodison Park when he signed him from Wigan Athletic during his time as Toffees boss, and is keen to be reunited with him at Manchester United.