Manchester City's top scorer pulled up hurt early in the second half of Saturday's 6-3 victory over Arsenal.
And, speaking at a press conference ahead of Tuesday's Capital One Cup tie at Leicester, Pellegrini confirmed Aguero will miss a Christmas programme of fixtures including the visit of Liverpool on Boxing Day.
Pressed on exactly how long the Argentine would be missing, he said: "Maybe the doctor can tell you, but at this moment we don't know how long it will be."
Aguero later 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!"
Aguero and summer signing Alvaro Negredo have struck immediate success as a partnership this season, scoring 20 Premier League goals between them.
Pellegrini has Edin Dzeko as back-up while Stevan Jovetic, whose last appearance came in the Capital One Cup in October, is available after injury.