Under the league's substance-abuse policy, the NFL has to make a decision on Gregory's status in the next 60 days.
Gregory was suspended all of last season for recurring issues involving the substance-abuse policy that date to the 2015 scouting combine, before he was picked in the second round by the Cowboys. He has been suspended four different times for failed or missed tests.
Gregory has admitted to issues with anxiety and marijuana use.
Under terms of the new collective bargaining agreement, players can no longer be suspended for a positive test for marijuana, but because he was penalized under the old system he would still need to be welcomed back by commissioner Roger Goodell. A player can continue to be suspended for missing tests or not taking part in a care plan.
The source said Gregory is doing well, which is why the reinstatement process was started.
Gregory, 27, last played in 2018, when he recorded 6.5 sacks in 14 games. He missed 30 of 32 regular-season games in 2016 and '17 because of suspensions. He has played in only 28 games in his career.
Last year, the Cowboys had their bid to sign Gregory to a contract extension denied by the league after it ruled players under suspension cannot be given new deals. Gregory remains under contract with the Cowboys since his contract has tolled. However, it is expected the team would sign him to a short-term extension.
Cowboys owner and general manager Jerry Jones has long backed Gregory and has favored changes in the substance-abuse program that deal more with care rather than banishment. Gregory was granted conditional reinstatement in the summer of 2018 before training camp, which allowed him to practice with the team, but he was suspended indefinitely a month after the season ended.
On the field, the Cowboys are in need of a pass rusher like Gregory after last year's sack leader, Robert Quinn, agreed to a deal with the Chicago Bears at the start of free agency. DeMarcus Lawrence, Tyrone Crawford, Dorance Armstrong, Joe Jackson and Jalen Jelks are the only defensive ends under contract. Jackson and Jelks, who spent the season on injured reserve, were rookies last season.