Lescott's contract expires at the end of the season and the England international appears unlikely to be offered a new deal by manager Manuel Pellegrini, having been limited to just 10 Premier League starts for City during the 2013-14 campaign.
Fellow Brits Les Ferdinand, Rob McDonald and Alan Walsh all spent time playing for Besiktas in the late 1980s and early 1990s and Lescott could become the latest in that line to sign for the Turkish giant.
"Lescott is still on our transfer wishlist," Ozen is quoted by The Sun as saying.
"Everything will become clear at the end of the season, players will be leaving and arriving, but I don't want to disclose too much information yet."
Lescott has made 160 appearances for City since joining the club in a big-money move from Everton in 2009. He has gone on to win the Premier League, FA Cup and Capital One Cup.