The Patriots will receive a 2021 seventh-round pick that could eventually improve to a sixth-round pick if conditions are met. In order for the seventh-round pick to turn into a sixth-rounder, the defensive end will have to be on Dallas' 53-man roster for four games the rest of this season, sources told ESPN's Todd Archer.
The NFL trade deadline is Tuesday.
Bennett, who turns 34 on Nov. 13, was suspended by the Patriots for last week's game for conduct detrimental to the team following a disagreement with defensive line coach Bret Bielema, which sources said occurred in front of other players.
"There's a lot of factors," Patriots coach Bill Belichick said Friday about the trade. "Multiple considerations on this. Look, in the end, I think Mike is a good player. We brought him here, he helped our team, I think he would help us. We only have a certain number of; certain things we have to work with and work around, and that's what we have to do."
Bennett, a three-time Pro Bowl selection who has 65.5 career sacks and 13 forced fumbles in 147 career games, was upset that his playing time with New England had declined sharply since he started the season opener. In Week 6, he played just 11 of 50 defensive snaps in a win over the New York Giants.
"It's America. You can voice your opinion about how you feel about certain situations. That's what I did," Bennett said after returning from his suspension. "... I got suspended. Lost money. What am I supposed to take away from that? There's no love lost. It's just how it is."
Asked if Bennett requested a trade, Belichick said, "I don't think that was a part of the conversation."
Bennett had 5 tackles, 2.5 sacks and 1 pass defended in six games with the Patriots. In that six-game span, he had been on the field for 133 of 374 defensive snaps (35.6%).
The Patriots clear about $2.5 million of salary-cap space by trading Bennett, although some of that will be used by the player the team signs to fill his roster spot. The figure is significant because the Patriots have been extremely tight to the cap, as they restructured the contract of starting right guard Shaq Mason to create space to acquire wide receiver Mohamed Sanu in a trade Tuesday. Without the restructure, the club wouldn't have been able to bring Sanu aboard.
That is the greatest benefit to the club, as the draft-pick compensation is modest. Bennett had become a niche player in the team's system, and had expressed his desire to play more, so he wasn't a great fit in New England despite high expectations after he had been acquired from the Eagles on March 14.
The Patriots had acquired Bennett and a seventh-round pick in exchange for a fifth-round pick, and then sweetened Bennett's contract through the 2020 season.
In a response to a tweet Thursday evening, Bennett posted, "I'm a @Patriots fan all day so it was cool while it lasted but never felt like it was going to work out."
It's the second straight year the Cowboys have made a bye-week trade. They acquired Amari Cooper from the Oakland Raiders and went on to win seven of their last nine games to win the NFC East last season.
Earlier Thursday, coach Jason Garrett said the team would be on the lookout for help, but he did not see a specific need to be filled.
"Excited about the group that we have. Really am," Garrett said. "And really focused on coaching the group that we have. There've been a lot of good things in the early part of the season we can build on and certainly plenty to correct and that's really what we're focused on as coaches more than anything else.
"Rather than thinking about, 'OK, getting somebody from somewhere else,' we'll let the personnel people address that here at the outset and if something comes to us, that's when we consider it."
The Cowboys put defensive end Tyrone Crawford on injured reserve last week with a hip issue that required surgery. Second-year defensive end Dorance Armstrong did not play a week ago because of a stinger but he was on the practice field this week. Robert Quinn suffered from rib muscle spasms that knocked him out of the second half of the win over the Eagles, but Garrett said the team anticipates he'll be able to practice when the Cowboys return from the bye week.
Bennett's brother, Martellus, was the Cowboys' second-round pick in 2008 and spent four seasons with the team before leaving in free agency.
The Cowboys and Patriots meet Nov. 24 at Gillette Stadium.
ESPN's Mike Reiss and Todd Archer contributed to this report.