Terim, who is also in charge of the Turkish national team, was voted out at a meeting of the club's board of directors after rejecting a contract extension.
"President Unal Aysal and chief executive Lutfi Aribogan have met with Roberto Mancini," Gala said 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 Premier League title success in 2012 and a 2011 FA Cup victory.
Galatasaray, who in January spent heavily on Didier Drogba and Wesley Sneijder, have made a disappointing start to the season, taking only seven points from five league games. They also suffered a 6-1 home thrashing by Real Madrid in the Champions League.