Speaking ahead of Glasgow's meeting with Cheetahs at Scotstoun, head coach Dave Rennie said: "He's certainly going to be out for the next few weeks we think.
"We'll have a bit more clarity by Tuesday, but he's unlikely for next week (Heineken Champions Cup quarter-final against Saracens).
"I think it's syndesmosis (high ankle sprain), so it's disappointing. He's defended really well at international level and I've been really excited for him, so it's a frustrating injury."
Earlier in the week assistant coach Kenny Murray provided an update on the progress of some of the other players on the injury list.
Speaking to The Offside Line, Murray said: "Ryan Wilson is not likely to be available in the next week or so (due to knee ligament damage) but is definitely looking to get back before the end of the season.
"Huw Jones is probably going to be another few weeks (having also suffered knee ligament damage). He is not back running yet, so I would say it is likely to be more around Ulster or Leinster time (at the start of April) that he will be available.
"George Horne had a shoulder knock which is why he was not in training as much with Scotland, and we are going to see how it is. We don't think it is anything major, just a wee tweak that we can manage a bit. So, he is doubtful this week, but I would hope he will be in contention for Saracens."
Murray added that Stuart Hogg is also targeting the Euro quarter-final clash against Saracens for his comeback from the shoulder injury he sustained whilst playing for Scotland against Ireland in the Six Nations at the start of February.
"He is back in training, he has not done a lot of contact so he will build that up this week going into next week," explained Murray. "He is not likely to be available for this week against the Cheetahs, but we are definitely not ruling him out of the Saracens game. We'll just see how he progresses towards the end of next week."