Hart's form is again under scrutiny after two errors in City's 3-1 Champions League hammering by Bayern Munich at the Etihad Stadium on Wednesday night.
The England number one failed to keep out a saveable Franck Ribery shot for the Germans' opener and was beaten too easily at his near post by Arjen Robben for the European champions' third.
They were the latest in a growing series of mistakes by Hart in an indifferent run now stretching back almost a year.
When asked if Hart could be dropped for City's home game against Everton in the Barclays Premier League this weekend, Pellegrini was equivocal.
Pellegrini said: "I will decide tomorrow if he starts. I don't tell you now if Joe will play tomorrow because the players don't know yet, but tomorrow I decide.
"Of course Joe knows he must improve the way he is playing but he has our trust."
When asked further Pellegrini, speaking at his pre-Everton press conference, said: "I am always contemplating different things. Tomorrow I will decide."
Hart has received the full support of England manager Roy Hodgson, who has said the keeper will play in the vital upcoming World Cup qualifiers against Montenegro and Poland.
Pellegrini is also supportive and is confident the 26-year-old will bounce back.