Cruz, 38, is hitting .270 with seven home runs and 22 RBIs in 35 games this season. He suffered the injury Sunday against Detroit.
If healthy, Cruz would have played his former team, the Seattle Mariners, this weekend. Cruz played in Seattle from 2015-18.
"I definitely want to play because I want to play every day, but definitely it's more special just to be able to come here and play against the old team," Cruz said, according to MLB.com. "It's something that you're always looking for."
The Twins recalled infielder Luis Arraez from Triple-A Rochester to take Cruz's roster spot.
Cruz is eligible to be activated May 24.