According to NFL Network's Jane Slater, Bryant apologized to the Cowboys coaching staff and "took full responsibility" after he showed up three hours late to practice due to a sinus infection and headaches.
Bryant also told Slater he felt he "should have informed coaching staff earlier" and that he was "fully prepared" for the consequences.
NFL Network's Ian Rapoport first reported Bryant was absent for the run.
The 28-year-old, who missed three games last season, is searching for a bounce-back campaign after he managed 50 catches for 796 yards and eight touchdowns during his first season with Dak Prescott at the helm.
Bryant is also eyeing his first 1,000-yard receiving season since 2014 after injuries limited him to 22 of a possible 32 appearances over the past two years.
The Cowboys will open training camp on July 24 in Oxnard, California.