The Scotland goalkeeper, 30, is one of several Ibrox players who have, following advice from PFA Scotland, rejected the chance to transfer their contracts from Rangers to Charles Green's newco. Steven Naismith has moved to Everton, Steven Whittaker has signed for Norwich and Jamie Ness has joined Stoke.
Northern Ireland duo Steve Davis and Kyle Lafferty are at Southampton and Swiss side Sion respectively, and McGregor could be next to leave Govan, with the Besiktas website stating that the club "began talks with goalkeeper Allan McGregor".
Despite Lafferty making his debut for Sion in a 1-1 friendly draw against Marseille on Sunday, none of the five players who have found new clubs have received international clearance as Rangers newco continue to dispute the players' status.
Green rejects the claim that the players are free agents and the former Sheffield United chief executive sent letters to clubs across the UK warning them that the players who objected to the switch to the newco were in breach of contract.