Mike played for the first 40 minutes of Friday night's 22-7 PRO12 win over Scarlets at Scotstoun.
The big prop is on the comeback trail after coming through a difficult couple of years of illness and injury problems.
Glasgow's remaining pool matches are on Sundays, so Euan, who has also picked up a minor niggle, can't play for religious reasons.
"He is not available for the Sunday games so in that position we felt we needed to bring someone in," said Gregor Townsend, the head coach. "Rossouw [De Klerk, the other registered tighthead] has been carrying an injury so we had to make the change.
"If we do make the quarter finals - we obviously need things to go in our favour - we can make three changes to our squad, which would be a good situation to be in. He [Euan] could be one of those three changes.
"Euan has been playing well - he made 14 tackles after playing just 40 minutes last weekend when the next best was 12 tackles in a full 80 - I'm very happy with how he has been playing but he's not available so we had to make the change."
It is still something of a gamble for Glasgow. Mike Cusack played 40 minutes against the Scarlets last weekend, his first professional action for almost two years. Though he scrummaged well enough against a young and inexperienced Scarlets front row, he was inevitably short of match fitness when it came to work in the open.
"The players and coaching staff are all delighted he is running around training and able to play," said Gregor. "There is still a bit of work for him to go in terms of getting back to, or even going beyond, his best level compared to how he was before, but he is ready for that.
"The tight is Mike's strength, though he is also a skilful player with good footwork and good hands. For a front row guy, though, it is all about scrum, tackle, contact area - that is 80 or 90 percent of their game. He also has an extra couple of days recovery, which is good in terms of him playing his first big game in a couple of years."