It's a five-year deal worth $43.75 million, sources told ESPN's Josina Anderson.
The deal includes $24 million in guaranteed money, according to sources.
Johnson was given the Cincinnati Bengals' franchise tag in 2013, sending his single-season pay skyrocketing above $11 million. It marked the second straight season that a Bengals player had been tagged.
After a productive year getting to quarterbacks in 2012, Johnson struggled doing the same in 2013, recording just 3.5 sacks. He had 11.5 sacks two seasons ago.
A 2009 third-round selection out of Georgia Tech, Johnson was signed to a four-year deal. Over the past few seasons with the Bengals, he helped bolster one of the league's stronger defensive lines.
When defensive tackle Geno Atkins signed a five-year, $55 million deal and fellow defensive end Carlos Dunlap signed a five-year, $40 million deal before last season, it appeared the Bengals would have difficulty retaining Johnson this offseason.
The Buccaneers also reached agreement with former Giants tight end Brandon Myers on a two-year contract with $4 million, sources said, and with former Seahawks defensive tackle Clinton McDonald on a four-year, $12 million deal, according to sources.
Myers, the Raiders' sixth-round pick in the 2009 draft, turns 29 in September.
After catching 79 passes in 2012 for the Raiders, Myers signed with the Giants hoping to be as productive or more with Eli Manning as his quarterback. Unfortunately, like a lot of other elements of the Giants' 2013 offense, Myers didn't click in New York. He caught 47 passes for 522 yards and four touchdowns and struggled to help as a blocker in pass protection or in the run game.
Myers' signature moment of 2013 was his inability to reel in a Manning pass that ended up being intercepted late in the October game in Chicago that dropped the Giants to 0-6 to start the season.
McDonald, 27, is one of the most underrated players on the Seahawks and his best NFL season in 2013.
He was a huge force in the interior line with 5.5 sacks, a fumble recovery and also had an interception.
The 6-foot-2, 297-pound McDonald didn't have a sack before last season. He made $592,941 in 2013.
ESPN.com Buccaneers reporter Pat Yasinskas, Bengals reporter Coley Harvey, Giants reporter Dan Graziano and Seahawks reporter Terry Blount contributed to this report.