Ticket sales have already topped 30,000 but Steyn is hoping the Scotstoun faithful turn out in huge numbers and fill the 60,000-capacity home of Celtic Football Club.
He told PA Sport: "We're hoping to get as close as we can to selling out Celtic Park.
"The 10,000 who go to Scotstoun are already making the same noise you get from 60,000 so it would be an amazing experience to see it filled with as many Glasgow fans as possible.
"Glasgow is a strong footballing city and we'd love to see it full of rugby supporters.
"I've never been to see a game at Parkhead. I've seen it from outside but it will be interesting to see if we get any time there this week.
"I'm not sure if any of the boys in the squad are Celtic supporters. I know Rory Hughes is a big Rangers man so it will be a strange one for him.
"A lot of the boys have spoken about games they've seen there and the atmosphere it produces. Hopefully we can use that to our advantage.
"The lure of playing at Celtic Park has been massive all season. We said going into the Edinburgh match in our final league game that if we won, we'd have the possibility of remaining in Glasgow for the rest of the season.
"It means we've not had to leave our own base, we've slept in our own beds. That could give us the edge we need."
However, South Africa-born centre Steyn believes there is still more to come from his side.
He said: "We definitely haven't peaked too early. There is still a hell of a lot from Friday's game that we can improve on despite the big scoreline.
"It's about keeping our feet on the ground and making the most of the learnings. Against Leinster, it's going to be a hell of a clash so we still have one more peak to climb."