A source told ESPN's Josina Anderson that the deal is expected to be for two years.
The Browns also went to the Alliance of American Football for quarterback depth, agreeing with Garrett Gilbert, the team said. The AAF, a developmental league, suspended all operations Tuesday.
Gilbert is an interesting signing given the quirky nature of the AAF. At a minimum he provides a needed third arm in OTAs and minicamp, and at best he might compete with Drew Stanton to be the backup to Baker Mayfield. Gilbert and Mayfield both attended Lake Travis High School in Austin, Texas.
The Browns had taken the position that Stanton would be a capable backup; Gilbert provides an option, if he works out. In eight games for Steve Spurrier's pass-happy offense in the AAF, Gilbert threw for 1,842 yards with 11 touchdowns. Orlando went 7-1 in his starts.
Gilbert's NFL career was not as successful. He was a 2014 sixth-round pick of the then-St. Louis Rams and spent time with New England, Detroit, Oakland and Carolina, where he played in one game before being released.
Burnett became a free agent when the Pittsburgh Steelers released him Monday, at his agent's request.
Last year the Steelers signed Burnett, who turned 30 in January, to a three-year, $14.25 million contract that included a $4.25 million signing bonus. Burnett played eight seasons for the Green Bay Packers before hitting free agency.
He missed four of the first six games of the 2018 season due to injury, and the Steelers started rookie Terrell Edmunds in his absence.
Burnett recorded 30 tackles and six pass deflections in a sub-package role over 11 games. Edmunds and third-year player Sean Davis occupied both starting safety spots for the season.
Burnett told NFL Network in the offseason that he wanted to be released because he wants to play safety instead of dime linebacker. The Steelers signed free agent Mark Barron as a starting linebacker who can also play Burnett's role in the dime package.
Burnett started all 102 games he played in for the Packers and had 9 interceptions, 44 passes defensed, 7.5 sacks and 717 tackles during his time in Green Bay.
ESPN's Jeremy Fowler contributed to this report.