The report states that the Lions are prepared to make a big-money move for the Arsenal man.
Galatasaray will however, have to wait until the turn of the year if they decide to sign the England international now as the Turkish transfer window deadline has passed.
Turkish Super League clubs cannot sign any new players until January.
Chamberlain is however, unlikely to part ways with Arsenal and still has another three-years left on his contract with the North London based club.
The 22-year-old has one goals and an assist in five appearances in all competitions for the Gunners this season.
The Lions won the Turkish league title for a record 20th time last season earning the right to wear a fourth-start on their crest.
The Istanbul giants booked a place in the Champions League group stages for a fourth consecutive season following their league and cup double.