The 30-year old Scotland goalkeeper is now a free agent after opting not to transfer his contract to Rangers' newco and is certain to move on this summer.
Reports late yesterday suggested that Besiktas were showing considerable interest with the clubs football committee chairman Ahmet Pipe stating last night that a deal was very close to being finalised.
"We wanted (Nice goalkeeper David) Ospina and were in negotiations for over a month," he said.
"However, the fees for the transfer kept changing and they were very high and we hardly had a chance to talk to him directly because he had so many representatives. In the circumstances, a transfer was too difficult.
"McGregor was next on our list of candidates and we had already negotiated with him. We have researched him and we know what house he wants, we have examined his lifestyle and his family structure. He's a very talented goalkeeper and we expect he will sign.
"We need a goalkeeper and we expect him to join our training camp in Austria on July 18."