McCoy completed an interview Monday, the Dolphins said. He's the first candidate to meet with the Dolphins since they were turned down Friday by former Titans coach Jeff Fisher, who chose instead to become coach of the St. Louis Rams.
Owner Stephen Ross sent his private jet to pick up McCoy in Denver to bring him to South Florida for the interview.
Meanwhile, the Oakland Raiders received permission to interview McCoy and Broncos defensive coordinator Dennis Allen for their head coaching vacancy.
McCoy was supposed to interview for the Jacksonville Jaguars' head coaching job last week, but with the Broncos then still in the playoffs, he never got to go and the position went to Mike Mularkey.
McCoy helped restructure the Broncos' offense midway through the season to capitalize on Tim Tebow's unique skill set, a move that buoyed the Broncos to the AFC West title.
Fisher turned down the Dolphins' offer and accepted the coaching job in St. Louis, the Rams confirmed Friday. Ross and general manager Jeff Ireland quickly hit the reset button in their search, but the rejection represented a major setback.
At one point Wednesday night, a source told ESPN NFL Insider Adam Schefter it was "80-20" that Fisher would choose Miami over St. Louis as his next job.
However, the Dolphins and Fisher could not agree on the structure of the organization, league sources said. Fisher was willing to make some compromises, but the Dolphins wanted Ireland to remain as the No. 1 man in the organization, and ultimately, it helped unseal any potential deal with the Dolphins, according to the sources.
Among the other candidates interviewed by the Dolphins, the only one with any NFL head coaching experience is Todd Bowles, thanks to his role as their interim coach for the final three games this season.
Other candidates interviewed by Miami have included Green Bay Packers offensive coordinator Joe Philbin, Cincinnati Bengals defensive coordinator Mike Zimmer and Chicago Bears special teams coordinator Dave Toub.
Information from The Associated Press was used in this report.