Fox Sports' Mike Garafolo reports that the deal is for three years.
McCown started 11 games for the Buccaneers in 2014, including their final eight. The 35-year-old threw for 2,206 yards, 11 touchdowns and 14 interceptions while completing 56.3 percent of his passes.
Brian Hoyer, who started 13 games for Cleveland last season, is set to become a free agent next month. The Browns' other quarterbacks include Johnny Manziel, who started two games in 2014 and entered a treatment facility in early February, and 23-year-old Connor Shaw.
The Browns will be the seventh team of McCown's career. He'll be entering his 13th NFL season in 2015.
- Molly Geary