Mancini was sacked as manager on Monday, with the club declaring the reason had been taken after City "failed to achieve any of its stated targets this year".
City failed to build on last season's Barclays Premier League title success and ended the season trophyless after Saturday's FA Cup final defeat against Wigan at Wembley.
Mills, who played for City between 2004 and 2009, told BBC Radio Five: "He will go down in Manchester City folklore for winning their first trophy in 35 years (the 2011 FA Cup).
"But only meeting one of the five targets, he's obviously failed on what he was asked to do.
"City's standing and expectations are completely different to those of a Reading or a Southampton."
Former City striker David White believes the club's fans will be disappointed with the Italian's departure, and feels early success for the new man is a must.
"I think overall they (the fans) were definitely behind the manager," he told Sky Sports News. "They have had success with him not seen over the past couple of years.
"The new manager comes in and if he starts well then you know things are quickly forgotten - it depends how the new manager starts next season."
And White feels Real Madrid coach Jose Mourinho, whose future at the Bernabeu is uncertain, would be a popular choice.
"Given the choice between Mourinho and (Malaga coach Manuel) Pellegrini, they (the fans) would love to see Mourinho at the club," White said. "But whether he fits in with what they want, he does tend to come and go at a football club. He would certainly be my choice and most City fans.
"It seems that they (the new manager) need to be successful in Europe and that is the aim of the club. They are looking at silverware every season.
"A lot of the players were incredible last year. They fell a little bit short this year. This makes a massive difference.
"We've got the top three teams that are going to finish all without a manager and if you want someone, you want to get get them early.
"It is difficult to keep that away from the press and then the manager."