The Turkish side are said to be keen after their poor start to the season.
It is being reported by Milliyet in Turkey that Gala are considering a move for the striker now he is a free agent.
The report claims that the poor form of current strikers Burak Yilmaz, Lukas Podolski and Umut Bulut has forced the hand of the Galatasaray board, who now hope to appease the fans with the signing of Adebayor.
Representatives for both parties have discussed a potential deal, as Gala look to overcome a poor start to the new season.
The club have won just once in the league this season and currently sit in eighth position.
Adebayor was finally released by Tottenham Hotspur at the weekend. The Togo striker had been told by Mauricio Pochettino that his services were no longer required at White Hart Lane – but failed to negotiate a move away from the club during the transfer window.
His release allows him to find a new club to begin his recovery from his Tottenham spell, but not a Premier League club. As Adebayor was still registered as a Tottenham player when the transfer window shut, he cannot join another Premier League team until January.
Which makes a potential switch to Galatasaray all the more likely.