The Argentina international hobbled off early in the second half at Etihad Stadium, having opened the scoring with his 13th league goal in 15 matches.
"Sergio has a calf injury," Pellegrini was quoted on the BBC ahead of Tuesday's League Cup quarter-final against Leicester City.
"The doctor can say how many weeks he will be out. I don't know at the moment. I think he will be out for at least one month."
Aguero has scored 48 league goals in 79 matches since joining City in 2011, and is second highest scorer behind Liverpool's Luis Suarez this season.
He has struck up a strong partnership with Alvaro Negredo since the Spaniards arrival from Sevilla, and his absence will give Edin Dzeko and Stevan Jovetic, who is returning from injury, an opportunity for a run in the side over the crowded Christmas period.
City's next league match is against Fulham on Sunday, while they have drawn Barcelona in the Champions League round of 16.
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