The former City boss is preparing his new side for the arrival of Chelsea in the first leg of their Champions League last-16 tie, and spoke exclusively to Sky Sports' Geoff Shreeves.
After dismissing suggestions that he still wants to be in charge at the Etihad Stadium, Mancini reckons that current manager Manuel Pellegrini cannot afford to slip up in their quest for trophies.
"It's part of the past. I worked really well in Manchester and I am proud of what we did, we changed the history of Manchester," he told Sky Sports.
"Now it's easy to win with Manchester City because they changed the mentality three years ago and they have good players - it's not difficult to win.
"I know football very well and when you decide to do this job and you should understand that if you don't win, they can sack you - sometimes without reason," he added.
"Man City will fight with Chelsea, I think. Chelsea have improved a lot in the last two months, but I think in the end City will win the title."
The Italian admits it will be a tricky task to overcome Jose Mourinho's side in Wednesday's clash in Istanbul and believes it was a 'miracle' that his team qualified from the group stage, having faced Real Madrid and Juventus.
"(Galatasaray) is an important club, like the level of the other top clubs. Maybe the league isn't as big as the others but I think it will improve in the next five years," he said.
"We performed a miracle. After the 6-1 (defeat to Real Madrid) here, it looked so impossible to get through. Instead, we had the chance and we played very well against Juventus and we went through.
"For us it's difficult, we know that Chelsea are used to playing these types of games. They've won the Champions and Europa League, and we know we don't have many chances to beat Chelsea but we want to do a good job," Mancini added.
"They have more of a chance than us, this is normal because they have really good players and they are a strong team but they should fight to win this game. In the end, probably, they will go into the quarter final but we will try to do our best."