Already handed a four-game suspension earlier in the offseason, Gregory would miss the first 10 games for at least his fourth misstep under the substance-abuse policy. It is not clear if Gregory failed another test or missed a test.
According to the policy rules, a missed test is treated the same as a failed test.
Gregory's representatives are still working on the appeal of his potential 10-game suspension and he has entered an undisclosed treatment facility and is not expected to report to training camp this week with the team, a league source told ESPN's Ed Werder. Whether he will participate at all is uncertain.
Cowboys coach Jason Garrett on Tuesday declined comment when asked about Gregory.
At last year's NFL scouting combine, Gregory tested positive for marijuana, which put him in the league's drug program. Considered a first-round prospect out of Nebraska, he fell to the second round, where the Cowboys selected him with the 60th overall pick.
The Cowboys and Gregory agreed that he would have a personal handler with him during the season to help him cope with issues related to anxiety.
Gregory signed a four-year deal worth roughly $3.8 million after the 2015 draft, including $1.4 million guaranteed. If he is suspended for 10 games, he would be out more than $350,000 of his $608,406 base salary this season.
Gregory played in 12 games as a rookie and did not record a sack. He suffered a high-ankle sprain in the season opener against the New York Giants and missed the next four games. Cowboys coaches credited him with 11 tackles and 16 quarterback pressures to go along with one tackle for loss.
The Cowboys will also be without DeMarcus Lawrence, who led the team in sacks last year with eight, because of a four-game suspension for violating the substance-abuse policy. Linebacker Rolando McClain has been suspended 10 games for violating the policy as well.
The Cowboys did not make huge additions to their defensive end group despite the suspensions. They signed Benson Mayowa as a restricted free agent to a three-year, $8.25 million deal and drafted Charles Tapper in the fourth round.
They have been reluctant to go after a veteran like Dwight Freeney, but their plans could change now if Gregory is to miss 10 games instead of four.
Gregory, as well as Lawrence and McClain, will be able to take part in training camp and preseason games. Gregory will also be able to work out on his own at the team's facility during the suspension, but he will not be allowed to practice.