The England defender is out of contract at the Etihad in the summer, and is not expected to be offered a new deal.
The Turkish club are hopeful of persuading the 31-year-old to sign a pre-contract agreement with them before the end of the season.
Lescott is set to turn 32 in August, and is likely to be a part of Roy Hodgson’s England squad for the summer’s World Cup, despite struggling for regular first team football at Manchester City this season.
He could go to the summer’s tournament with his future undecided, should Beskitas not manage to get a pre-contract agreement sorted in time.
West Ham showed an interest in signing the former Everton and Wolves defender in January, but a bid fell through, with Lescott looking to demand more as a free agent in the summer.
There has been a recent explosion of money in Turkish football, and Besiktas are reportedly willing to offer a contract worth £100,000 a week for three years.