The Rams will receive a mid-round draft pick in exchange for Quinn, and the teams will swap late-round picks, NFL Network Insider Ian Rapoport reported.
No trades can become official until the new league year starts on March 14, but teams can agree to trades.
The Rams earlier reached an agreement to acquire cornerback Marcus Peters from Kansas City. ESPN's Adam Schefter reported the Rams wanted to include Quinn in the deal with the Chiefs, but Kansas City received second and fourth-round draft picks instead.
The 27-year-old Quinn has played his entire career for the Rams. He started 14 games in 2017, when he had 8.5 sacks.
Quinn was selected to the Pro Bowl in 2013 and 2014, and has 62.5 sacks in his seven-year NFL career. His best season was 2013, when he had 19 sacks and was selected first-team All-Pro.
Quinn is under contract for two more years and is scheduled to make $11.4 million in 2018 and $13 million in 2019. He was notified of the trade in a conference call with the Dolphins, Rapoport reported.