Sources told ESPN's Jake Trotter that Goodson's deal is for one year.
The Browns were on the hunt for a veteran linebacker after cutting Christian Kirksey and letting Joe Schobert walk in free agency. Kirksey signed with the Green Bay Packers, and Schobert agreed to a five-year deal with the Jacksonville Jaguars. The Browns' top two linebackers going into the offseason were 2019 rookies Mack Wilson and Sione Takitaki.
NFL Network first reported the Goodson deal.
Goodson, who was acquired by the Packers in a Week 1 trade with the New York Giants last year, was the base inside linebacker next to Blake Martinez in run-stopping defenses. He played in all but one game, with nine starts, and recorded 45 tackles, one quarterback hit and one pass breakup.
Goodson, who turns 27 on May 29, was a fourth-round pick (No. 109 overall) of the Giants in the 2016 draft and played in 37 games (with 20 starts) in three seasons for New York.
ESPN's Rob Demovsky contributed to this report.