Aguero, who flew to Barcelona for consultation with Dr Ramon Cugat last week, went under the knife on Monday to address cartilage damage.
"Recovering from an arthroscopy on my knee", he tweeted.
After that he was unable to train fully in the build-up to the second leg of City's Champions League quarter-final against Liverpool on April 10.
City's season finishes on May 13 away at Southampton, while Argentina's first game at the World Cup is against Iceland on June 16.
City secured the title after local rivals Manchester United suffered a shock 1-0 defeat at home to the Premier League's bottom club West Brom.
City also claimed the title on the final day of the 2013/14 season, but Aguero believes his third league winner's medal in England is just reward for the best football he has been involved in since joining from Atletico Madrid in 2011.
Aguero missed Saturday night's win at Tottenham after suffering an injury setback, but manager Pep Guardiola said last week he remained hopeful the 29-year-old would be available for the "last games" of the season.
Returning to full fitness in time for this summer's World Cup will now be Aguero's main concern, with the showpiece event in Russian Federation now just two months away.
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