Cingrani was placed on the 10-day disabled list on April 21 with a strained right oblique. In five appearances this year, he had allowed one run over 4 2/3 innings with a 1.07 WHIP and a .222 opponents' average. He pitched in 65 games in 2016.
Price said Cingrani is still experiencing some soreness.
"He is able to maintain his arm strength and range of motion," Price said. "However, not to the point yet where we would clear him to get up on the mound and start throwing [bullpen sessions]. We really want to focus on making sure the soreness is gone."
The Reds are currently carrying one left-hander in their bullpen, Wandy Peralta.
Cincinnati has dealt with oblique injuries over the past couple of seasons, including those to pitchers Tim Adleman and Rasiel Iglesias and outfielder Billy Hamilton. Those experiences give the Reds' trainers a guide on how to manage Cingrani's recovery.
"It's a challenging injury to deal with, and the last thing we want to do is have him come back and reinjure it and have to start over, so we will err on the side of caution," Price said. "But he has progressed."
Nick Travesio, on the 60-day disabled list while recovering from offseason shoulder surgery, has felt comfortable while playing catch, but he has not yet started throwing off a mound, Price said. The right-hander is the No. 21 prospect in the Reds' organization, according to MLBPipeline.com.
Andy Call is a contributor to MLB.com based in Cincinnati who covered the Reds on Friday.