City's leading scorer this season with 19 goals injured his calf during Saturday's 6-3 victory over Premier League leaders Arsenal.
Pellegrini confirmed Aguero's absence at a news conference on Monday.
City are due to play Championship side Leicester City in the League Cup quarter-finals on Tuesday.
Aguero Tweeted: "It's confirmed - my calf injury will leave me out for at least a month. The good news is recovery work starts today. We started with therapeutic massages on the area and some bike work. I'll keep you posted. Thanks for the support."
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