The Italian has been out of work since losing his position at Manchester City at the end of last season.
Speculation over a move to Turkey began shortly after Fatih Terim's departure earlier this week.
And Galatasaray confirmed on their official Twitter feed that talks between Mancini and the club president have taken place.
A short statement, accompanied by a picture of Mancini alongside Galatasaray's chief executive, read: "President Unal Aysal and chief executive Lutfi Aribogan have met with Roberto Mancini."
Terim, who remains manager of the Turkish national team, was sacked after refusing the offer of a contract extension.