He'll need to play at least a few rehab games with the Bisons before returning to the Blue Jays' lineup.
Kevin Pillar and Anthony Gose have been sharing duties in center field in his absence.
Rasmus is hitting .222 with nine home runs and 19 RBIs in 37 games this season, though he'd turned around a slow start and was hitting .289 with four homers and 10 RBIs in the 10 games prior to his injury.
In other injury news, right-hander Sergio Santos threw a bullpen session on Sunday and has since departed for his rehab assignment with Double-A New Hampshire. The 30-year-old reliever hasn't pitched since May 9 when he started to experience right forearm tenderness.
Santos is 0-2 with five saves and a 9.00 ERA in 12 innings with the Blue Jays this season.
He could rejoin the team as early as this weekend, when Toronto travels to Baltimore for a four-game series against the Orioles.
Jamie Ross is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.