According to the Mirror, the Istanbul side sacked Terim last week after he rejected 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, the report said.
The club posted on Twitter that President Unal Aysal and chief executive Lutfi Aribogan had met with Roberto Mancini, attaching a picture of the Italian alongside Aribogan.
Mancini 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, the report added.
The 48-year-old has also won trophies in Italy with Inter Milan, including three successive league titles from 2006 to 2008, the report further said.
Galatasaray, who in January spent heavily on the likes Didier Drogba and Wesley Sneijder, have had a disappointing start to the season, according to the report.
They have taken only seven points from five league games, and suffered a 6-1 home thrashing at the hands of Real Madrid in the Champions League, the report mentioned.
--ANI (Posted on 30-09-2013)