The French forward went down clutching his leg after a challenge in the ninth minute of the 1-1 draw with Borussia Dortmund in a pre-season friendly in Chicago.
Nkunku, who was signed for close to 60 million pounds ($76.32 million), came off midway through the first half in discomfort.
"Nkunku has sustained a knee injury which will rule the forward out for an extended period," the club said in a statement.
"Nkunku has undergone an operation and will now begin a rehabilitation programme with the club's medical department."
The injury is a blow to Chelsea's plans of returning to the top four this season under new manager Mauricio Pochettino after the London club finished 12th in the 2022-23 campaign.
Nkunku scored 23 times for RB Leipzig across all competitions last season.