The 29-year-old sustained a medial ligament tear against Harlequins and saw a specialist on Tuesday afternoon to assess the state of the injury.
Cipriani, who won the last of his 14 international caps against France in August 2015, will now miss at least the first two rounds of the European Champions Cup as well as several Premiership fixtures, starting with Sunday's match against the defending champions Exeter Chiefs.
Wasps also revealed that back-rower Alex Rieder needs surgery after dislocating his shoulder against Quins, and will be out for four months.
"It's obviously not ideal to be without some of our key players, but it is likely to happen to a number of clubs at some stage during the season," Wasps director of rugby Dai Young told the club website.
"Unfortunately, we've picked up a few injuries early doors which is going to stretch us, but we just need to deal with that. We have a strong squad and that's what building quality throughout the squad is for."
Cipriani returned to Wasps in 2016 after six years away from the club, playing first for the Melbourne Rebels in Australia before returning to England with Sale Sharks.