Reyes was placed on the 10-day DL with a right lat strain, which he suffered during Wednesday's start against the Milwaukee Brewers.
Cardinals general manager Michael Girsch said Thursday it was a "significant" strain.
"It is not going to be a couple starts," Girsch said. "It's going to be more than a few starts. We don't have all the information yet so we don't know beyond that how long it's going to be. But it is not a minor injury and we're gathering more information."
Reyes pitched four scoreless innings on Wednesday in his return from Tommy John surgery, allowing three hits with two strikeouts and two walks.
He threw 73 pitches (42 for strikes) but fatigue was noticeable for Reyes, and the Cardinals and manager Mike Matheny noted "a drop in velocity."
Reyes was 4-1 with a 1.57 ERA in 2016, striking out 52 in 46 innings. He suffered his elbow injury in spring training last year.
In other moves Wednesday, the Cardinals recalled left-hander Austin Gomber, right-hander Mike Mayers and first baseman Luke Voit from Triple-A Memphis and sent outfielder Tyler O'Neill and right-hander John Gant to Triple-A.