Ashcraft, 25, was struck in the leg by a comebacker in the third inning of Thursday's loss to the Los Angeles Dodgers.
He took the loss in that game, falling to 3-5 with a 6.78 ERA through 13 starts this season.
The Reds recalled right-hander Ricky Karcher from Triple-A Louisville to fill Ashcraft's spot on the active roster.
Karcher, 25, was a 13th-round draft pick by Cincinnati in 2017 and has not yet made his major league debut.
He is 2-2 with a 9.27 ERA and one save in 22 relief appearances this season at Louisville.