Kompany lasted just nine minutes against the Black Cats after aggravating a calf injury that has dogged him over the past few seasons, and Pellegrini said on Tuesday that he did not know exactly when the Belgian would return.
"At this moment it is difficult to tell you an update about that," the manager told reporters.
"Vincent is doing a lot of different things to try to understand and find the reasons why he has so many injuries.
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"Maybe not so soon but for the last part of the season he could be fit."
City conceded just one Premier League goal with Kompany on the pitch but have kept only one clean sheet in the 11 matches he has missed.
Midfielder David Silva has urged his team mates to step up in the central defender's absence.
"He is a key player for us and our captain. With or without him we have to work hard to concede less. It's a duty we have to fulfil before too long," he said.
"Any team will miss their captain when they're out and we're no exception. When you look all over the field there are lots of figures willing to step up to the plate when needed."
City are third in the Premier League table and take on 11th-placed Everton later on Wednesday.
(Reporting by Simon Jennings in Bengaluru, editing by Nick Mulvenney)
