Speaking to BBC Scotland, head coach Gregor Townsend said: "We don't expect him to play in the first two or three games of the season.
"We'll just see what happens over the next few weeks."
Finn was stretchered off in Galway after a collision with team-mate Zander Fagerson. He was kept in hospital for over a week for observation, but is now back in light training.
"He's working on skills, he's very keen to be involved," Gregor continued.
"Everything has gone well so far with his recovery, but obviously we're being very cautious on when he can come back to training with the team and also full training.
"So far his progress has been really good."