The St. Louis Post-Dispatch reported Friday that Fisher was putting the final touches on a contract with the Rams.
"I'm really excited that the Rams are the best fit for me," Fisher said in an interview with the newspaper. "I'm looking forward to coming to work."
A statement from the Dolphins said there had been discussions with Fisher about their vacant head coaching position.
Fisher left the Tennessee Titans following the 2010 season. He spent 17 years with the club, posting a regular-season record of 142-120 with four division titles and six playoff appearances.
The Titans reached the Super Bowl at the end of the 1999 season, but lost to St. Louis.