Bruce was scratched from the starting lineup Sunday because of a sore knee but pinch-hit and struck out during a 4-3, 10-inning win over Milwaukee. He had tests Sunday night that detected the tear.
The Reds have been hit hard by injuries since spring training. They're currently missing starters Mat Latos and Tony Cingrani, catcher Devin Mesoraco, closer Aroldis Chapman and now their cleanup hitter.
Sale threw 40 pitches before Monday's game against the Chicago Cubs at Wrigley Field. General Manager Rick Hahn said the left-hander will go on a rehabilitation assignment in the "coming couple weeks."
Sale has been sidelined since April 17, when he threw 127 pitches in a 3-1 loss to the Boston Red Sox. He was sore the next day and went on the 15-day disabled list on April 22.
Giambi was removed from Saturday night's game against the White Sox after grounding into a double play in the fourth inning. Giambi began the season on the DL after breaking a rib during training camp and the 43-year-old is 0 for 10 this season.