Elliott wrote he has "worked hard to make better decisions and to live up to the high standards that are expected of me" but added he "need[s] to work harder to ensure I do not put myself in compromised situations in the future."
ESPN's Todd Archer reported Goodell planned to see Elliott regarding an incident in May at a music festival in Las Vegas.
According to TMZ Sports, Elliott pushed a security guard after getting into an argument with his girlfriend. Las Vegas police briefly handcuffed Elliott but didn't take him into custody.
Cowboys owner Jerry Jones spoke in support of Elliott shortly after the incident.
"I think that the main thing is that I don't see anything that needs supporting," Jones said, per Archer. "In terms of his status with us, [it] has not been impacted in any way. And frankly, I know how conscientious he has been in the offseason, and that's good enough. No, I don't see that having any consequences for us."
NFL Network's Mike Garafolo reported Jones made sure the Cowboys' team counsel was with Elliott during Tuesday's meeting with Goodell.
Goodell issued a six-game suspension to Elliott in August 2017 for a violation of the NFL's personal conduct policy after the league investigated domestic violence allegations against the two-time Pro Bowler.
Archer noted the NFL's personal conduct policy allows for Goodell to levy punishments against players regardless of whether they were formally charged with a crime.