Quinn, who visited The Star last week and also met with the New Orleans Saints, is getting a new one-year deal worth $8 million from the Cowboys and he can earn $1 million more in incentives, according to sources.
Quinn was set to make $11.8 million in the last year of his contract. Earlier in the month, the Dolphins paid Quinn a $1.12 million roster bonus and reportedly were willing to pay some of Quinn's base salary to facilitate a trade. The Dolphins recently paid $5 million of the $7 million guaranteed to quarterback Ryan Tannehill as part of a trade to the Tennessee Titans.
The Dolphins, however, are not paying any of the money in Quinn's new deal with the Cowboys, the sources said.
The Cowboys have a need to improve their pass rush.
At the league meetings in Arizona earlier this week, Cowboys executive vice president Stephen Jones said the discussions with DeMarcus Lawrence on a long-term deal have reached an "impasse," but the club remains hopeful a contract can be worked out before the July 15 deadline. Lawrence, who had 10.5 sacks last season, was given the franchise tag for the second straight year, guaranteeing him at least $20.5 million in 2019.
Randy Gregory, whose six sacks in 2018 were second on the team to Lawrence, has been suspended indefinitely for again violating the NFL's substance abuse policy and the terms of his conditional reinstatement. While the Cowboys hope Gregory will be able to return in 2019, there are no assurances.
Taco Charlton, Dallas' first-round pick in 2017, is coming off shoulder surgery and had just one sack last season. Tyrone Crawford had 5.5 sacks last season, but the NFL is looking into an incident at a Florida bar recently that could lead to a penalty through the personal conduct policy.
Quinn, who turns 29 in May, had 6.5 sacks last season after he was dealt by the Los Angeles Rams to the Dolphins. For his career, he has 69 sacks, including a career-high 19 in 2013. In 2015, he signed a four-year, $57 million deal with the Rams after two straight Pro Bowl seasons in which he recorded 29.5 sacks.
He has 24 sacks over the past four seasons.
The Cowboys are without a first-round pick in the upcoming draft because of last season's trade for wide receiver Amari Cooper from the Oakland Raiders. The Dolphins are in a rebuild mode with new coach Brian Flores and general manager Chris Grier.
The deal also follows the Cowboys' approach in free agency, making short-term, cost-effective signings with the likes of Randall Cobb, George Iloka, Christian Covington and Kerry Hyder that will allow them to take the best players available in the draft in Rounds 2-7.
The Dolphins will need to rebuild their defensive line, and edge rusher is arguably their biggest need. Miami lost veteran defensive end Cameron Wake in free agency to the Titans. Disappointing 2017 first-round pick Charles Harris is currently the Dolphins' top edge rusher.
ESPN's Cameron Wolfe contributed to this report.