Bergamasco missed his country's defeats to Argentina and Australia over the past two weekends and is also a doubt for the Azzurri's Six Nations opener against Ireland on February 5.
"It is necessary to intervene surgically to assess and repair the injury," said Italy's chief medical officer Roberto Pozzoni.
"The recovery time will be between 60 and 120 days, but we will only have the full picture with regard to his full recovery after the surgery."
With over 50 European appearances to his name, Bergamasco played in both of Stade Francais' opening Amlin Challenge Cup games against Crociati Rugby and Leeds Carnegie this season, but will miss the next four matches at least.
Stade Francais sit five points clear at the top of Pool 4 and line up against Bucharest Oaks in back-to-back fixtures in December.