City announced the Italian's departure in a statement on their official website, claiming the club had "failed to achieve any of its stated targets this season" a year after winning the Premier League title.
However, White feels that employing Mourinho would be a good way for the club's owners to get the fans back on side.
"I think overall [the fans] were definitely behind the manager. They have had success with him not seen over the past couple of years," White told Sky Sports News. "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."
"Given the choice between Mourinho and Pellegrini, [the fans] would love to see Mourinho at the club. 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 [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 them early. It is difficult to keep that away from the press and then the manager."
Mancini took over at the Etihad in December 2009.