The Bengals received a seventh-round 2018 NFL Draft pick in the deal.
Cincinnati announced the move on Tuesday. Flowers, 25, originally entered the NFL as a sixth-round pick by the Bengals in the 2014 NFL Draft. The Bengals also placed running back Cedric Peerman on injured reserve Tuesday after he suffered a shoulder injury in the team's third preseason game Sunday against the Washington Redskins at FedEx Field.
Flowers has mainly worked on special teams during his three-year NFL tenure. The University of Arizona product started one game at linebacker in 2014. He recorded 13 total tackles and a passes defended that season. Flowers was on injured reserve in 2015 with a shoulder injury. Flowers tallied one fumble recovery last season for the Bengals.
Cincinnati made the trade despite having questions at the position following the five-game suspension of middle linebacker Vontaze Burfict. Flowers is due $690,000 this season, the final year of his rookie contract.
New England signed three rookie free agents on Monday, including linebackers Christian Kuntz and Nick Usher. The Patriots also signed defensive lineman Michael Bart. New England released former Bengals tight end Matt Lengel and former Minnesota Vikings defensive end Caleb Kidder on Sunday.