Sebastián Cancelliere and Scott Cummings were left at home after both picked up knee injuries in the round of 16 tie at Kingston Park.
Warriors, who will be based in Stellenbosch, face Stormers in Cape Town on Friday evening and Bulls in Pretoria seven days later.
Peter Murchie, Glasgow’s assistant coach, expects it to be a tough examination.
“Both teams are very good sides and have been going well post-Christmas,” he said. “All the South African teams are picking up results. It’s going to be a big test for us.
“Stormers are a good team, the Bulls are a good team so it’s a good challenge for us. We’re going into the business end of the season now with a lot of away games which is a good challenge for us as a team. It’s time that we get to work.”
Glasgow are third in the United Rugby Championship on 50 points, while the Stormers are three points behind in fifth place and the Bulls are eighth on 43 points.
The top eight qualify for the play-off but you need to be in the top four to land a home tie. With three rounds of fixtures remaining it’s all to play for and Murchie believes Williamson can step up.
“Max is a player who has been in the environment for quite a long time in terms of training,” said the coach. “He played for the Scotland under-20s in the recent Six Nations so it’s a good opportunity for him to come into the group and he’ll be ready to take an opportunity.
“He’s one of many players we see with a bright future so it’s good that we can spend two weeks with him.
“It’s a good opportunity for guys to get tight as a group. He’ll learn from it but he’s there to be ready to play as well.”
Glasgow Warriors squad
Forwards: Simon Berghan, Jamie Bhatti, Fraser Brown, Rory Darge, Jack Dempsey, Zander Fagerson, Tom Gordon, Richie Gray, Rob Harley, Oli Kebble, Johnny Matthews, Nathan McBeth, Kiran McDonald, Ally Miller, George Turner, Murphy Walker, Max Williamson, Ryan Wilson.
Backs: Jamie Dobie, Robbie Fergusson, Cole Forbes, Sam Johnson, Sean Kennedy, Stafford McDowall, Josh McKay, Domingo Miotti, Ali Price, Ollie Smith, Kyle Steyn, Ross Thomson, Sione Tuipulotu, Duncan Weir.