Perez suffered the injury in Friday night's loss to the Oakland Athletics.
"Juan's not a young guy," Blue Jays manager John Gibbons said as the team awaited news on the extent of the 34-year-old's injury. "He's a warrior. He's a great competitor and he's done a good job for us. It's unfortunate. We don't know how it's going to end up. You feel for anybody in this business."
Perez pitched an inning and two-thirds in Toronto's loss to the Oakland Athletics before feeling something go wrong.
"Any time the guy calls you out there, yeah, it's usually not good," Gibbons said.
The Blue Jays recalled righty reliever Mickey Storey from their triple-A affiliate, the Buffalo Bisons. Storey, 27, appeared in two games this season, pitching 3 2/3 innings with a 7.36 ERA.
"He can fit that role that we need: a guy that can pitch some innings, two, three four, maybe," Gibbons said. "A strike-thrower. He's been here before. Kind of long guy, yeah. We can use him in different roles. Hopefully we don't get to that point, but if we do, he can throw multiple innings for us."
Perez made 19 appearances this season, going 1-3 with a 3.69 ERA. Spending most of his professional career in the minor leagues, Perez has also been with the Pittsburgh Pirates, Philadelphia Phillies and Milwaukee Brewers.
The Blue Jays got a strong contribution early in Perez's tenure in Toronto, as he didn't allow an earned run in his first 14 appearances.