Cusiter was assessed by Julie McBirnie, a shoulder specialist, at Spire Murrayfield Hospital in Edinburgh and does not require an operation.
Head coach Gregor Townsend told www.glasgowwarriors.org: "Everyone at Glasgow Warriors is disappointed that a quality player like Chris Cusiter is likely to be out of action until around the middle of December.
"It's encouraging, though, that he doesn't need surgery, and we know he's in the best possible hands with our medical team. Chris is a tough competitor, and he'll be working hard to return to fitness and make a significant contribution to our campaign."
Scotland head coach Andy Robinson said: "It's disappointing for Chris because he has made such a good start to the season. I wish him a speedy recovery."