The former England international picked up the injury during the team's win over Perpignan in a Top 14 match last month.
Laporte told reporters: "Andrew is out for a month and possibly forever. He has always had a problem between his neck and his shoulder. The surgeon has told him that it's not necessary to have a new operation.
"He will now have a month of complete rest and if the pain doesn't go away, then he will have to stop playing rugby."
The 34-year-old spent the majority of his career with Sale Sharks before moving to the French club in 2012.