Beachy, recovering from elbow-ligament replacement surgery in June last year, made his fifth minor league rehab start on Thursday for Triple-A Gwinnett at Rochester.
After giving up three runs, two earned, three hits and four walks in three innings, Beachy was examined by a Braves doctor in Atlanta on Friday.
"We think it's a very minor setback," Wren said. "We're not sure exactly when we'll insert him back in. It's really a function of when he's feeling good again.
"He was a little tender after the last start and this time he was a little more tender. ... They feel like it's just inflammation they have to get out of there."
Beachy is 2-1 with a 3.27 ERA in five minor league rehab starts.
Wren said the team remains confident "everything is going to be fine" with Beachy's comeback.
"He's disappointed," Wren said. "I think he was excited to get back out there and perform. So I think he's disappointed in that regard, but he's also a smart young guy. He knows that's the best thing."
Earlier Friday, manager Fredi Gonzalez said Beachy was on track to pitch in Tuesday's doubleheader against the New York Mets.
Wren said the Braves would select a replacement starter for Beachy on Sunday.
Beachy was perhaps Atlanta's top starter when his 2012 season ended, leading the NL with a 2.00 ERA. He was 7-3 with a 3.68 ERA in 2011.