The Italian has been looking for a club after parting company with Premier League outfit Manchester City in May, while Galatasaray have moved quickly in their search for Fatih Terim's successor.
Terim, who recently took up the post of Turkey national team boss, was dismissed by Gala earlier in the week after a five-game winless streak.
And now Mancini, who began his managerial career at Serie A outfit Fiorentina before a highly successful spell with Inter, is the most likely to take over having flown out to Turkey to undergo talks.
The 48-year-old most recently spent four years with City, and his tenure was widely regarded as a success among supporters, after winning the FA Cup in 2011 with a 1-0 victory over Stoke City.
They then dramatically won the 2011-12 Premier League title when a late Sergio Aguero goal gave the side a 3-2 victory over QPR on the final day of the season that saw City pip rivals Manchester United to the title on goal difference.
But he paid the price for surrendering the title back to United last season, and also losing the FA Cup final 1-0 to Wigan Athletic in a season starved of silverware.
