The Tigers said Sanchez has a right pectoralis major strain and made the DL move involving him retroactive to Aug. 9. Soria has a left oblique strain, the team said.
Sanchez could miss three- to-four weeks, the team said.
Sanchez left Friday night's game against the Toronto Blue Jays in the fifth inning with a strained muscle in his right side. Soria left Saturday's game after straining a muscle in his left side while warming up before the 10th inning.
Robbie Ray will be called up to start Wednesday's game against the Pittsburgh Pirates in place of Sanchez, who is 8-5 with a 3.53 ERA this season.
Soria, who was acquired from the Texas Rangers in a trade this season, is 2-4 with a 3.58 ERA and 17 saves (all with Texas).
Soria is 1-1 with a 10.38 ERA in six games with Detroit.
The Tigers did not make any moves to replace Sanchez and Soria, leaving their bullpen short for Sunday's series finale against the Blue Jays. Closer Joe Nathan has appeared in both games at Toronto so far, getting his sixth blown save Saturday. Detroit will add two players before Monday's game at Pittsburgh.
The Tigers lead the AL Central by 1½ games over the Kansas City Royals entering Sunday's games.
The Associated Press contributed to this report.