Webb, who missed the Six Nations because of the injury, is set to leave Ospreys for Toulon at the end of the season.
That switch will make him ineligible to play for Wales, the most recent of the scrum-half's 33 caps coming in December against South Africa.
A statement from Ospreys' medical team read: "Rhys pulled out of Wales' Six Nations campaign and went on to have surgery on an ongoing knee problem.
"He will be unavailable for the remainder of this season and his focus is now on getting his recovery right for next season."