Flaherty underwent imaging tests on his right shoulder Tuesday.
Reyes, a 27-year-old right-hander, was scheduled to receive what manager Oliver Marmol termed a therapeutic injection on Wednesday at the Andrews Institute in Gulf Breeze, Florida.
The Cardinals were barred by Major League Baseball from contacting their players during the lockout and learned of the injuries when the players reported.
Flaherty, 26, was 11-8 with a 2.75 ERA in 2019, went 4-3 with a 4.91 during the pandemic-shortened 2020 season and 9-2 with a 3.22 in 17 starts last year, when he was slowed by an oblique injury,
"Once he started to experience it, he shut it down," Marmol said. "Now it gives us a chance to actually have [a] shot to get him back."
The oft-injured Reyes had 29 saves last season. He is expected back at camp on Friday.
"I would say with Reyes, you're looking at that being really tough for him to break camp with the team," Marmol said. "That would be pushing it."
All-Star catcher Yadier Molina is absent because of what the Cardinals say are personal reasons. Marmol doesn't know when the 39-year-old will arrive.
"I wanted to give him some space," Marmol said. "We're talking about one of the most competitive humans on earth. If he's not here, it's for a reason, and I want to make sure I'm sensitive to that and give him the space."
Reliever Giovanny Gallegos, who replaced Reyes as the Cardinals' primary closer late last season, was expected to arrive at camp on Thursday. He, too, was delayed because of personal reasons.
Reliever Genesis Cabrera arrived after flight delays.