The news was first reported by the Washington Post.
Poe, 27, signed a one-year, $8 million contract with the Falcons last year and quickly set the tone as a run-stuffer. He also showed flashes of pressuring the quarterback, although that was not his specialty.
He started all 16 games in 2017 and was credited with 39 tackles, 4 tackles for losses, 2.5 sacks and 10 quarterback hits in 745 defensive snaps. The Falcons included weight-based incentives into his deal, adding bonuses if he weighed 330 pounds or less during the season, which he met.
The Falcons also installed their goal-line "Poe Package," lining the burly veteran at fullback against the Dallas Cowboys. During his time with the Kansas City Chiefs, Poe had a pair of 1-yard rushing touchdowns and a 2-yard touchdown toss off a jump pass in eight offensive snaps.
Said the Falcons' Ricardo Allen: "That's the definition of brotherhood right there. No matter what they call on Poe to do, he's going to go out there and give them his all."
Poe, the 11th overall pick in the 2012 NFL draft, spent his first five seasons with the Chiefs. A two-time Pro Bowl selection, he has 15.5 career sacks, 240 tackles and 2 forced fumbles.
ESPN's David Newton and Vaughn McClure contributed to this report.