The Patriots will send a seventh-round draft pick to the Bengals in the deal, which was announced Tuesday.
A 2014 sixth-round draft choice out of Arizona, Flowers has been mostly a special-teams player for the Bengals. He took only two snaps on defense in the final game of last season, but was a core special-teams player the entire year.
Flowers was viewed to be on the roster bubble in 2017, and with the Bengals trading him for a late-round pick, it is an indication they felt the same way.
A potential tipping point for the Bengals might have come recently in practice, when the 6-foot-3, 245-pound Flowers was yelled at by coach Marvin Lewis after almost injuring one of the team's quarterbacks.
As for why the Patriots would target Flowers, they have been hit by injuries to linebackers who are significant contributors on special teams, with Shea McClellin, Geneo Grissom and rookie Harvey Langi among those sidelined.
With the season opener Sept. 7 against the Kansas City Chiefs, who feature two of the league's most dynamic return men in Tyreek Hill and De'Anthony Thomas, special teams will be a major part of the Patriots' game plan.
To make room on the roster, the Patriots released LB Christian Kuntz, who was signed Monday.