The Argentina international picked up the injury in Wednesday's 5-1 win over Tottenham and looks certain miss the first leg of the UEFA Champions League last-16 tie with Barcelona on February 18.
City manager Manuel Pellegrini will also be without Aguero for the meetings with Chelsea in both the Premier League and the FA Cup, and the former Atletico Madrid man faces a race against time to be fit for the League Cup final against Sunderland on March 2.
"The doctor has seen Aguero and he will be out for a month," Pellegrini said on Friday.
"It will be difficult because he is a top player for us but we have the players to replace him.
"He has just come back from injury and Alvaro Negredo, Stevan Jovetic and Edin Dzeko can replace him."
Aguero has been in fine form this season, scoring 26 goals in all competitions.
