The Padres also signed right-hander Vince Velasquez to a minor league contract and recalled right-hander Shaun Anderson from Triple-A El Paso.
Snell, 28, injured his groin in the first inning of Sunday's 8-0 loss to the Los Angeles Dodgers. After retiring his first two batters, Snell was facing Trea Turner when he suddenly left the mound with a 2-1 count. He was removed from the game and limped slightly as he walked off the field.
Snell, the 2018 AL Cy Young Award winner, is 7-6 with a 4.22 ERA.
Velasquez, who was released by the Philadelphia Phillies on Tuesday, is expected to be added to San Diego's active 28-man roster this weekend in St. Louis and make a start during the three-game series against the Cardinals, the Padres said.
A seven-year veteran, Velasquez, 29, is 3-6 with a 5.95 ERA this season.