"He's in counseling working really hard," Dee Haslam said. "It truly is up to him. "We have high expectations for our players."
Hunt is suspended for the first eight games this season for violating the NFL's personal conduct policy, stemming from a February 2018 incident in which he shoved and kicked a woman during an altercation outside his downtown Cleveland residence. Release of the video from the incident prompted the Kansas City Chiefs to release him last November.
Hunt signed with the Browns in February to a one-year deal worth up to $1.1 million with no money guaranteed.
Both the Haslams and general manager John Dorsey met with Hunt before signing him.
"You have to take the situation very, very, very seriously," Dee Haslam said. "We spent a lot of time and John spent a lot of time with Kareem and felt like he has potential as a person."
Jimmy Haslam said "talks are ongoing" with the league to get approval for Hunt to stay with the Browns during the suspension.
"Kareem understands what he has to do to continue being part of the Cleveland Browns," he said.