Watson picked up the injury while playing for Emerging Scotland against a Tonga XI at Scotstoun a fortnight ago.
Watson was one of Glasgow's top performers in pre-season and earned praise from Franco Smitn after a man-of-the-match display against Northampton.
Asked about teh extent of Watson's injury, Smith told Scotland Rugby News: "It's negative. I don't think we'll see him for 12 to 14 weeks. I was really disappointed with that injury. Obviously, it's hard to lose anybody, especially if you already had a plan or the plan was in place to use him. I think the disappointment is not just with us, it's obviously with him. He was going to get this opportunity, but he will. Another opportunity will arise. Like I said, we've built the squad already in the last seven games."
Another of Glasgow's young players, Kerr Johnston, starts on the wing. Johnston made his debut as a replacement almost nine months ago, also in Wales, against the Dragons and Smith says he deserves another opportunity.
"It feels like nine months ago, but he's been training and playing well for us," said Smith. "The A-team games that he's been playing, he's part of the plan, the progression.
"If we have so many players in that role of national interest and some top players there, then the opportunities are few, but it becomes more as the confidence grows. I'm excited that he's one of the young boys that we're looking to keep for a longer time. Good to see him get a run out this week."