Llovera was placed on the 15-day injured list with a Grade 2 right flexor strain Saturday.
Llovera exited Friday's win over the Brewers after throwing 1.2 relief innings, and he'll be forced to miss at least two weeks after being diagnosed with a Grade 2 strain. A timetable for the right-hander's return isn't yet clear, but he isn't eligible to be activated until at least the end of July.