Laporte damaged his cartilage and lateral meniscus in a challenge with Brighton defender Adam Webster.
The operation was successful but City have not yet given an indication of how long the 25-year-old will be out.
Laporte made 35 league appearances last season with Bernardo Silva the only City outfield player to play more.
The injury ruled him out of a possible debut for France after he had been called up by coach Didier Deschamps for the Euro 2020 qualifiers against Albania and Andorra.
Laporte’s absence leaves City with just Nicolas Otamendi and John Stones as central defenders following the departure of Vincent Kompany to become player-manager of Anderlecht in the summer.