The Broncos will get Dawson and the Patriots' seventh-round pick in 2020. In return, the Patriots will get Denver's sixth-round pick in 2020, sources told ESPN's Adam Schefter.
Trading Dawson, a 2018 second-round pick, breaks up a logjam at one of Patriots' deepest positions on the roster.
NFL Network first reported the Dawson trade.
The Patriots thought highly of Dawson, trading up to select him with the 56th overall pick in the 2018 draft. But Dawson didn't appear in a game in his rookie season, opening on injured reserve because of a hamstring injury and then never breaking through onto the 46-man game-day roster after being activated from IR in mid-November.
The 5-foot-10, 198-pound Dawson played primarily in the slot and also worked at safety at times. Four-year veteran Jonathan Jones, who helped successfully cover Kansas City Chiefs receiver Tyreek Hill in the AFC Championship Game, is the Patriots' top slot cornerback.
The Patriots also drafted cornerback Joejuan Williams in the second round of the 2019 draft, further adding to a position that includes starters Stephon Gilmore and Jason McCourty, as well as a future potential starter, second-year player J.C. Jackson.
In college, Dawson played in 48 career games for Florida, posting 82 tackles, six interceptions and 23 passes defended.
The Broncos had high marks on Dawson in the 2018 draft and, after selecting wide receiver Courtland Sutton in the second round (No. 40), he was a player they would have strongly considered if available.
Depth in the secondary has been a concern throughout training camp in Denver. One of the Broncos' starters, free-agent signee Bryce Callahan, has not played in the preseason while being given some days off in training camp.
Callahan fractured his foot last December with the Bears and coach Vic Fangio has consistently said they were being "prudent.''
The addition of Dawson also allows the Broncos to play Kareem Jackson at safety instead of corner, which is the team's preference.
The 27-year-old Bodine, a fourth-round pick by the Bengals in 2014, is the third offensive lineman acquired by the Patriots via trade this week.
He started 10 games at center for the Bills in 2018 -- his first with the team. His position on the 53-man roster was jeopardized when the Bills made Mitch Morse the highest-paid center in the NFL this offseason.
Buffalo traded for lineman Ryan Bates shortly after Bodine suffered an oblique strain in the team's first preseason game, pushing Bodine further down the depth chart. With Morse's full clearance from concussion protocol as Saturday's final roster deadline approaches, Bills general manager Brandon Beane made the rare trade with a division rival.
For the Patriots, starting center David Andrews is likely out for the season after being diagnosed with blood clots in his lungs following the third preseason game, so the Patriots have bumped four-year veteran Ted Karras into the starting spot.
Karras has five career starts since joining the Patriots as a 2016 sixth-round pick, and Bodine now gives them a more experienced option, as he has started 74 career games between the Bengals (2014-2017) and Bills.
The Bills have now acquired two sixth-round picks and a fifth-round pick in the 2020 draft over the past two days; they sent OL Wyatt Teller and a 2021 seventh-rounder to the Cleveland Browns on Thursday in exchange for fifth- and sixth-round selections.
ESPN's Jeff Legwold and Marcel Louis-Jacques contributed to this report.