The deal will be worth $21 million in fully guaranteed money at signing, the source said, and up to $36 million with potential incentives and bonuses.
Lawson, a 2016 first-round pick, was an early target to upgrade a Dolphins pass rush that produced a league-worst 23 sacks in 2019. He played for Dolphins defensive line coach Marion Hobby at Clemson.
The Bills declined to exercise Lawson's fifth-year option last offseason. General manager Brandon Beane issued a challenge to Lawson, urging him to rise to the occasion.
Lawson, who turns 26 in June, answered with the best season of his career, recording a career-high 6.5 sacks and 13 tackles for a loss. One of the final remnants of the pre-Beane/Sean McDermott era, Lawson has given the GM and head coach credit for helping him mature over the past three seasons.
He has 16.5 sacks, 5 forced fumbles and 25 tackles for loss in four NFL seasons.
ESPN's Marcel Louis-Jacques contributed to this report.