The team recalled Xavier Scruggs from Triple A Memphis to take Jay's place on the roster.
Jay also has a bone bruise on his left thumb and sat out the last three games. He underwent surgery last October on his wrist and has yet to be 100% this season, according to the Cardinals.
"It's one of those things where it's still early in the season, and we're just trying to stay ahead of it now," Jay said. "Obviously, I want to be out there playing, but I think this is the best thing for now. Hopefully, we'll get back on track, and I'll get back out there as soon as I can."
Scruggs was hitting .229 with six home runs at Triple A. Jay is hitting .248 with six RBIs and a .330 on-base percentage in 30 games this season.
St. Louis hosts the Detroit Tigers on Friday. The Cardinals are currently in first place in the NL Central.
- Molly Geary