The extent of the injury was discovered during a magnetic resonance imaging exam performed in Seattle.
The right-hander exited Tuesday's game against the Mariners early after yielding a two-run double to Raul Ibanez in the third inning. He threw just 42 pitches in allowing six runs for the second consecutive outing.
Peavy has gone 6-4 with a 4.30 ERA in 11 starts, surrendering 63 hits over 67 innings while walking 15 and striking out 66.
His spot in the rotation will be taken by left-hander Hector Santiago, who has compiled a 1-4 record with a 3.40 ERA in 15 games (five starts) this season.
Celebrity Families of 2014 [PHOTOS]