Reyes, ranked as the No. 42 overall prospect in baseball by ESPN's Keith Law, is 0-1 with a 15.00 ERA in four appearances this season.
The right-hander has struggled with control this season as a setup man for St. Louis, walking six batters in three innings.
Reyes, 24, missed most of the past two seasons after having Tommy John surgery in 2017 and surgery to reattach a tendon in his lat after making just one start in 2018.
"I think we're all seeing similar things," manager Mike Shildt told reporters, according to the St. Louis Post-Dispatch. "It's been a while since he's been able to get into consistent competition, and we're asking him to do it at the big-league level in high-leverage situations. And ultimately that's the decision I make.
"A lot of confidence in Alex. We'll always have confidence in Alex. He's a special young man, with special stuff and skill sets. Still working around where he's more consistent with his pitches. Obviously all the pitches are there. Just a matter of being more consistent."
In other moves Sunday, the Cardinals also optioned infielder Yairo Munoz to Memphis. Right-hander Giovanny Gallegos and left-hander Tyler Webb were recalled from Triple-A.