Steyn had already been a doubt for Scotland's final Six Nations match against Ireland, but he picked up a separate issue while in Dublin. The winger has sustained a minor stress fracture in his ankle and will wear a protective boot for three weeks, with his return to action expected to take several additional weeks after that.
The timing is far from ideal, as Glasgow head into a crucial phase of their season.
Steyn, who has been nominated for the Six Nations Player of the Tournament award, left head coach Franco Smith with mixed feelings.
"He's got a little stress fracture. That will keep him in a boot for three weeks and then we will try and bring him back. We can't put a specific time frame on it. It's always difficult with those things. Unfortunately, he'll be missing some of the games now.
"It was sustained in the Ireland game. There was an open wound going into that game after he fell on Dupont's leg the week before and he got cut. But this one is maybe a little bit of something that happened in the week leading up to it or obviously in the game.
"Obviously, I'm super proud of him [for the nomination] and the way he's gone about his business. He dealt well with the disappointment of not being selected for the Lions tour last year. I think he had his hopes up for that.
"Since then, he's made the international arena his own. He knew the hard work that we put in here and the way we coach and develop the players here is going to assist him in doing that. Across all the Scottish games this year, we could see how hard the boys have worked and mentally grown here in the club environment.
"I'm really proud of him and his diligence and his hard work and his belief but also the mental toughness that he's shown that makes him the leader that we want. We, as a club, are really proud to see him being nominated for that role."