The England goalkeeper has been criticised for a number of uncertain performances for club and country over the last year, and even England coach Roy Hodgson has expressed concerns over his form.
Hart had another uncomfortable game as City were surprisingly beaten at promoted Cardiff last week, although Pellegrini insisted he still had "trust" in his current first-choice keeper.
Pellegrini said: "For the moment Joe will continue being our goalkeeper but I spoke with him and he knows he must be quickly in his top form again.
"I am sure he is a top goalkeeper and will do it again. I said from the beginning my teams never play with names, they play with performances.
"Joe knows perfectly he needs good performances in the next games. At the moment he has all the trust."
When asked if Hart could be dropped, Pellegrini added: "I think it is not the moment.
"That doesn't mean we don't have another goalkeeper. We have Costel Pantilimon. But we know Joe has had a difficult moment and those players need trust in those moments.
"I saw the whole of last season, but last season is past. We must see about the present and future and for this moment, I think it is best for the team that Joe Hart continues.
"Joe is working with another goalkeeping coach. I trust a lot in Xabier Mancisidor, he is a very good goalkeeping coach. I worked a lot with him in Malaga and Real Madrid and I am sure Joe will improve in a lot of things during the whole year.
"Joe is a very young goalkeeper, I think he arrived to the top in England maybe too young and he needs a lot of things to improve and Xabier will do it."