A source told ESPN's Adam Schefter that Hunt signed a one-year contract. He will be a restricted free agent after the contract expires, meaning the Browns will control his rights after next season.
The Kansas City Chiefs released Hunt in November after a video surfaced that showed him shoving and kicking a woman last February at his residence in Cleveland.
He neither was arrested nor faced charges for the February incident. He also was accused of punching a man in June.
The NFL placed Hunt on the commissioner's exempt list shortly before he was released. He is not eligible to play until the NFL completes its investigation into the incidents and a decision on potential discipline is made.
A source close to the situation told ESPN's Dan Graziano that the discipline process was "still ongoing and nowhere near done." A source told Graziano that Hunt met with NFL investigators in January as part of the process.
Hunt, now that he has signed, will again be placed on the commissioner's exempt list until the NFL makes a final decision on discipline.
Browns general manager John Dorsey drafted Hunt for the Chiefs in 2017, when Dorsey was the team's GM.
"I want everybody to know we have done extensive research in regards to this case, this player," Dorsey told reporters during media availability Monday. "He understands and takes full responsibility for the egregious act he committed. He is extremely remorseful for his actions."
Dorsey said he talked to many people about Hunt, but did not talk to the woman, nor did he talk to groups that deal with domestic violence or violence against women. He said he did not reach out to the woman.
Hunt, 23, apologized to the woman in the February incident, the Chiefs organization and his family in an interview with ESPN after his release.
He apologized again in a Browns statement on Monday.
"First off, I would like to once again apologize for my actions last year. What I did was wrong and inexcusable. That is not the man I was raised to be, and I've learned a great deal from that experience and certainly should have been more truthful about it after the fact. I'm extremely grateful that John Dorsey, Dee and Jimmy Haslam and the Cleveland Browns organization are granting me the opportunity to earn their trust and represent their organization in the best way possible on and off the field," he said.
"I am committed to following the necessary steps to learn and to be a better and healthier person from this situation. I also understand the expectations that the Browns have clearly laid out and that I have to earn my way back to the NFL. I'm a work in progress as a person, but I'm committed to taking advantage of the support systems that I have in place to become the best and healthier version of myself."
"I am committed to following the necessary steps to learn and to be a better and healthier person from this situation. I also understand the expectations that the Browns have clearly laid out and that I have to earn my way back to the NFL."Kareem Hunt
Dorsey said Hunt voluntarily began counseling shortly after he was released by the Chiefs and "is now working toward being a better man going forward."
He added that Hunt's signing is "by no means ... a guarantee of anything. As we all know, trust has to be earned. That has to be earned with the Cleveland Browns organization and the community of Cleveland moving forward. This will be a day-to-day thing with earning trust."
Asked if the Browns would have a zero tolerance policy with regard to Hunt, Dorsey replied, "You know what, I'll say yes."
Dorsey said he expects the NFL's investigation to wrap up in a couple of weeks and he hopes Hunt is present April 1 for the start of offseason workouts.
Hunt led the NFL in rushing in 2017 as a rookie with 1,327 yards and eight touchdowns in helping Kansas City qualify for the playoffs. He also had 53 receptions for 455 yards and three touchdowns. Last season, he rushed for 824 yards and seven touchdowns in 11 games and had seven receiving touchdowns before he was released.
"I know this, if you talked to anybody who's been in the locker room with Kareem Hunt, they'll tell you he was a really good teammate," Dorsey later told a handful of beat reporters.
Hunt's signing is a bit of a surprise given the Browns have a talented corps of backs. Nick Chubb ran for 996 yards in nine starts as a rookie, and averaged 5.2 yards per carry. The team also has Duke Johnson, who is a very accomplished change-of-pace back and receiver out of the backfield. Johnson also averaged 5 yards per carry (on 40 carries) in 2018, and in 2017 had 1,041 total yards and seven touchdowns. The Browns signed Johnson to a three-year, $15.6 million contract last June. He is under contract with the Browns through the 2021 season.
ESPN's Pat McManamon contributed to this report.