Swisher left Sunday's game against Texas after hurting himself. Cleveland hosts Cincinnati on Monday night.
The injury is another setback in a rocky season for Swisher. He was on the disabled list in June with a sore knee and is batting .212 with eight homers and 42 RBIs.
Swisher struck out in all three plate appearances Sunday and was taken out for a pinch-hitter in the ninth inning. He was wearing a brace on his wrist after the game.
The switch-hitting Swisher batted under .200 for most of the first three months. He has made nine errors at first base and has primarily been the DH for several weeks.
Swisher signed a four-year, $56 million contract with Cleveland after the 2012 season. He batted .246 with 22 homers and 63 RBIs last year.