The contract is expected to be a five-year, $61 million pact with at least $30 million guaranteed.
He intercepted Russell Wilson's pass at the goal line with 20 seconds left in the game, preventing a go-ahead touchdown from the Seattle Seahawks.
Thus ends one of the most interesting careers with the New England Patriots.
Butler's durability was impressive in his time with the Patriots, as he played 98.8 percent of the defensive snaps in 2015, followed by 96.7 and 97.8 the next two seasons.
Butler, 28, was mysteriously benched by Bill Belichick in Super Bowl LII and many second-guessed the decision following the Patriots' 41-33 loss to the Eagles. Butler later denied any off-field misconduct, saying he never missed a curfew or do anything to hurt the Patriots' chances. There's another $3 million via incentives.More news: Appeal against Khaleda's bail goes to full bench tomorrow
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New England also signed Stephon Gilmore a year ago, signaling Butler was falling out of favor with Bill Belichick.
The Titans will add Butler to a secondary already featuring Ryan and Adoree Jackson, who started 16 games after being drafted 18th overall.
Henry served as Murray's backup for two seasons, although he outrushed the veteran back by 85 yards on eight fewer carries in 2017, then ran for 156 yards in the Titans' playoff victory over the Kansas City Chiefs. He's also been an important asset as a receiver, making 85 receptions the past three seasons (30 games) while catching almost 92 percent of his targets from Tom Brady last fall. Though injuries limited him to just 14 regular season games in his first two seasons in Foxborough, Lewis's impact as a dynamic threat out of the backfield helped keep New England atop the National Football League hierarchy.
AP Sports Writer Brett Martel in New Orleans and AP Sports Writer Kyle Hightower in Boston contributed to this report.