Perez left in the seventh inning of Friday's loss to Oakland after throwing 56 pitches in 1 2/3 innings. He flexed his arm in pain, bringing manager John Gibbons and trainer George Poulis to the mound.
Perez, 34, is 1-2 with a 3.69 ERA in 19 games with Toronto this season. He also pitched for Pittsburgh, Philadelphia and Milwaukee.
This is Storey's second stint with the Blue Jays this season. He made two relief appearances in May, going 0-0 with a 7.36 ERA.
Also Saturday, the Blue Jays activated right-hander Drew Hutchison (elbow) from his rehab assignment and optioned him to Buffalo.