Such a lay-off could make the Argentina striker doubtful for the glamour Champions League tie against Barcelona in February.
Pellegrini had previously said he expected Aguero, who suffered a calf injury in Saturday's 6-3 thrashing of Arsenal, to be out for at least a month.
There remains uncertainty as to the true extent of the problem but it could be worse than first feared.
Pellegrini, speaking after City's 3-1 Capital One Cup quarter-final win at Leicester, said: "I am not a doctor. It is very difficult.
"The doctor has said he has at least one month or eight weeks. We will see.
"I don't how many weeks but it will be at least one month. It is very difficult to know at this moment how many weeks it can be."
Aguero's absence is an obvious blow to City with the 25-year-old having been in outstanding form this season, with 19 goals to his name.
He had found the target again against Arsenal but then pulled up when under no pressure early in the second half.
Aguero is certain to miss the Christmas programme, which includes a home clash with Liverpool in the Barclays Premier League, and could also be out for both Capital One Cup semi-final legs in January.