The source added, however, that the league will closely monitor and support Tunsil in the event intervention is needed.
A video posted on social media minutes before the start of the NFL draft last Thursday night showed Tunsil smoking from a bong through a gas mask. The Dolphins and Tunsil confirmed the video was from two years ago, when the offensive lineman was a college player at Ole Miss. Tunsil's draft stock plummeted Thursday night until Miami selected him with the No. 13 overall pick.
"I do not have a drug problem," Tunsil said to the Miami media after getting drafted. "You can check all my college tests. I never failed one."
There is a behavior clause in the collective bargaining agreement that would make it possible for Tunsil to enter Stage 1 of the program.
The clause states: "Behavior [including but not limited to an arrest or conduct related to an alleged misuse of substances of abuse occurring up to two football seasons prior to the player's applicable scouting combine] which, in the judgment of the medical director, exhibits physical, behavioral, or psychological signs or symptoms of misuse of substances of abuse."
Brian McCarthy, the NFL's vice president of communications, would not comment on Tunsil's eligibility for the program when contacted by ESPN.
"We cannot comment on the status of any player due to the confidential nature of the NFL-NFLPA substance of abuse program," McCarthy said.
Most of Tunsil's transgressions are issues that Ole Miss now must deal with, including his recent admission that he took money from a Rebels coach.
The Dolphins did an extensive background check on Tunsil during the pre-draft process and were comfortable with his character and the selection. Tunsil was the No. 2 overall player on Miami's draft board, according to the team.
"I think a lot of work is done for months and months, and one [thing] two hours before the draft is totally somebody out to get a guy," Dolphins owner Stephen Ross said. "It's not a question of this guy changed overnight. The work that we've done on him, the people we've talked to and even subsequently talking to, we're going to continue that.
"Meeting the kid, you know that this is a good kid. It's going to be a great choice."
Tunsil said he is coming to Miami with a lot to prove and will do what it takes to fit in well with his new NFL team.
"Don't question my character," Tunsil said. "You can't judge a book by its cover. I'm a good person. I'm a laid-back guy. I don't do much."
ESPN's Kevin Seifert contributed to this report.