Fatih Terim was sacked as Gala coach earlier this week after rejecting a contract extension. Terim, who is also in charge of the Turkish national team, was voted out at a meeting of the club's board of directors.
"President Unal Aysal and chief executive Lutfi Aribogan have met with Roberto Mancini," Gala posted on Twitter, attaching a picture of the Italian alongside Aribogan.
Mancini, 48, has been out of work since being dismissed as City manager in May after a three-and-a-half-year reign at the Etihad Stadium which included a Barclays Premier League title success in 2012 and 2011 FA Cup glory.
Galatasaray - who in January spent heavily on the likes Didier Drogba and Wesley Sneijder - have had a disappointing start to the season, taking only seven points from five league games, while they suffered a 6-1 home thrashing at the hands of Real Madrid in the Champions League.