Hart was not at his best on Sunday, beaten twice from corners as City fell to a 3-2 defeat against newly promoted Cardiff City.
And while Pellegrini has stated that Hart remains his first-choice goalkeeper, he warned this could change if Hart's form continues to falter.
"For the moment Joe will continue being our goalkeeper but I spoke with him and he knows he must be on his top form again quickly," Pellegrini said.
"I said from the beginning my teams never play with names, they play with performance. Joe knows he needs good performances in the next games.
"We know Joe has a difficult moment but players need trust in those moments. At the moment he has all the trust."
Pellegrini believes the England number one can continue to improve with the help of goalkeeping coach Xabi Mancisidor.
"Joe is a young goalkeeper," he continued.
"He reached the top in England here maybe too young and he needs a lot of things to improve. Xabi will do it."
