This is not a new topic.
Elliott, the fourth player taken in the 2016 draft, enmeshed himself in controversy when he pulled down a woman's top, exposing her breast, while they stood on a float in a parade last weekend.
No charges have been filed, and the woman has not publicly complained about Elliott's actions.
Still, those actions show a lack of judgment, considering the NFL has yet to close its investigation into an alleged domestic-violence incident last summer involving Elliott and a former girlfriend.
Understand, a player doesn’t have to commit a crime to earn a suspension from commissioner Roger Goodell. All he has to do is violate the league's player-conduct policy.
Talk to enough people in the Cowboys organization and it's clear they hope their message of self-awareness resonates with Elliott before he finds himself in a meeting with Goodell.
When you ask members of the organization about Elliott’s actions, there's an exasperation to their tone and an edge to their text messages. They don’t think Elliott is being malicious, but the club wants the 21-year-old to make better decisions.
More than once, Elliott has been lectured about the importance of understanding he’s going to be scrutinized every time he leaves home. The Cowboys knew he liked to party when they drafted him out of Ohio State, and they have talked to him about how to party safely, whether it’s using security or arriving through private entrances.
The messages, they say, are always received well by Elliott, but his actions don’t show consistent change.
This team is built around Elliott’s ability every bit as much as the Cowboys' championship teams of the 1990s were built around Emmitt Smith. Elliott makes the offensive line better, the receivers better and quarterback Dak Prescott better.
The Cowboys can’t afford to have him suspended because the commissioner believes he’s bringing negative attention to the NFL.
Coach Jason Garrett had a conversation with Elliott in August, when the rookie visited a marijuana dispensary the day of a preseason game against Seattle. The visit wasn't illegal, but it was a poor look for one of the NFL's more visible players.
"I don't think it was a good decision," Garrett said at the time. "He and I talked about that. It was just a poor decision on his part and young players sometimes just have to understand that perception can be reality.
"You have to understand you're under a microscope 24 hours a day.”
Last month, Elliott was at a bar in Columbus, Ohio, when one of his closest friends was arrested for trying to take a gun into the establishment. That’s not Elliott's fault, but it brought more negative attention.
Elliott led the NFL in rushing in 2016, gaining 1,631 yards with 15 touchdowns as the Cowboys went 13-3 and won the NFC East.
He's a star, and he has essentially been a star since his senior year of high school. The Cowboys say he also has some of the same entitlement issues that would plague any 21-year-old millionaire.
The Cowboys vow to remain consistent and patient in their approach to help Elliott improve as a player and a person.
More than once, Garrett has said he understands players make mistakes. He wants them to accept responsibility for those mistakes, be accountable and take action to rectify the issues.
Garrett will tell you establishing a team's culture is a daily endeavor. The process for getting Elliott to develop some self-awareness is the same.