The deal is worth $33.75 million with $24 million guaranteed, a source tells ESPN's Adam Schefter.
Johnson will fill a major hole in the Browns defense at safety, where Cleveland juggled several players in and out of the lineup last season. He joins a rotation that will now include Ronnie Harrison, who eventually won a starting job after coming over to Cleveland in a trade from Jacksonville before last season, and Grant Delpit, a former second-round pick who missed his entire rookie year in 2020 with an Achilles injury suffered in training camp.
Johnson, who has served as a cornerstone of the Rams secondary since he was inserted into the starting lineup in Week 5 of his rookie season in 2017, served as the defensive signal-caller on the NFL's top-ranked defense in 2020.
Voted as a team captain, Johnson, who turned 25 last December, intercepted a pass and had eight passes defended this past season.
Johnson's signature play with the Rams came in the NFC Championship Game during the 2018 season, when he intercepted a pass from New Orleans Saints quarterback Drew Brees in overtime that set up a game-winning field goal to advance to Super Bowl LIII.
He was on pace for a standout 2019 season, intercepting two passes in the first three games, but he suffered a shoulder injury in Week 6 that sent him to injured reserve for the remainder of the season.
In four seasons, Johnson has 8 interceptions, 32 pass deflections and 1 forced fumble.
ESPN's Lindsey Thiry contributed to this report.