The 56-cap Scotland lock was operated on by surgeon Umberto Fazzi at Ross Hall Hospital in Glasgow following his team's RaboDirect Pro12 final defeat to Leinster.
But he is now a doubt for the start of next season after opting to have the long-standing injury fixed.
The 32-year-old was left out of Vern Cotter's first Scotland squads for matches against Canada, United States, Argentina and South Africa.
Glasgow say a number of other players are currently being assessed by the club's medical team following Saturday's 34-12 defeat in Dublin.
A Warriors spokesman added: "Glasgow Warriors would like to thank the Leinster team doctor Professor John Ryan and all the staff in the A&E department of St. Vincent's Hospital in Dublin for all their help following the game."