Eric Galko of Optimum Scouting reported the news Wednesday, though he noted it is "far from a lock."
Gordon, 26, is scheduled to get out of a rehabilitation center Thursday. The 2013 Pro Bowler has been out of the NFL for the last two seasons due to substance abuse issues. He has played in five total games since his breakout 2013 campaign, during which he led the NFL in receiving and broke a number of yardage records.
The league denied Gordon's appeal for reinstatement in May. In August, NFL Commissioner Roger Goodell said Gordon's reinstatement was "not under active consideration," per Mary Kay Cabot of Cleveland.com.
"A lot gets focused on Josh the player but we are arm and arm with the league," Browns executive vice president Sashi Brown told Cabot. "The most important thing for Josh is for Josh as a young man to get himself to a point where he gets past some of the issues that have prevented him from being able to participate and once we get to that stage, those questions will be answered in terms of his return to the field.
"But we care first and foremost in the league about the young man and that's where the focus will remain until we get to that point."
NFL spokesman Brian McCarthy recently told Pro Football Talk's Mike Florio that the league still had no update on Gordon's status.
If the NFL does reinstate Gordon, the Browns retain his rights.