The 29-year-old has not started a match since a 1-0 league victory over Chelsea on March 4. He suffered the injury during a training session and aggravated the problem in the Manchester derby after a challenge by United's Ashley Young.
His last appearance came from the bench against Liverpool in the Champions League quarter-final, second-leg last week when the striker claimed he "couldn't run" after 24 minutes on the pitch.
Earlier on Tuesday, Aguero wrote on his Twitter account: "Recovering from an arthroscopy on my knee. Fully motivated to get back soon to the field."
With the league title wrapped up and City's Champions League campaign ended by Liverpool, the club can cope without their all-time leading scorer, who has 199 goals for the Manchester Club after scoring 30 in all competitions this season.
However, the striker is expected to be a key figure for Argentina, who play their opening World Cup match against Iceland on June 16.
(Reporting by Hardik Vyas in Bengaluru; Editing by Jon Boyle and Toby Davis)
