Fisher, 53, had been a top candidate for the vacant jobs in both St. Louis and Miami.
St. Louis was believed to be the more attractive destination due to the presence of young quarterback Sam Bradford, combined with the Rams having significant salary cap space, while owning the second overall pick in the upcoming NFL Draft.
The Miami Herald reported last week Ross was prepared to top any offer tabled for Fisher's services.
Fisher parted ways with the Titans last January after more than 16 seasons, during which he established Tennessee as a regular playoff contender and took the 1999 team to the Super Bowl, losing 23-16 to the Rams.
Fisher was 142-120 in 16 seasons with the Houston Oilers/Tennessee Titans franchise. His teams were 5-6 in the playoffs.