The in-form Argentine striker limped out of Saturday's 6-3 victory over Arsenal during the second half.
Aguero later confirmed on Twitter that he would be missing for 'at least a month'.
Pellegrini also hinted that his fellow South American could be back early in the New Year.
The Chilean tactician has, however, now revealed that he cannot put a timescale on Aguero's recovery.
Were he to be out of action for the 'eight weeks' which are being suggested, he would be a major doubt for City's crucial clash with Barcelona in the last-16 of the UEFA Champions League in February.
Pellegrini said after watching his side overcome Leicester City 3-1 to reach the semi-finals of the Capital One Cup: "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."