Foley was en route to Colorado on Tuesday afternoon, a source said.
McElwain is 22-16 in three seasons at Colorado State, including 10-2 this year. The Rams were one of two Group of 5 teams with victories this season over two Power 5 conference teams, defeating Colorado and Boston College.
Named the Mountain West Conference's coach of the year on Tuesday, McElwain has turned the Rams into an offensive powerhouse. Colorado State averages 498 yards per game (13th best in the nation), while boasting the nation's second most efficient quarterback in Garrett Grayson and the top receiver by yards per game (149.1) and touchdowns (17) in Rashard Higgins.
Before taking over Colorado State, McElwain, a Montana native, was Alabama's offensive coordinator/quarterbacks coach from 2008 through 2011.
The Gators are looking for a replacement for Will Muschamp, who was fired last month. The Gators went 6-5 this season and a program that grew accustomed to putting points on the board ranks 90th this season in total offense, which is Florida's best mark during Muschamp's tenure.
New England Patriots offensive coordinator Josh McDaniels, a rumored candidate for the job opening, downplayed that talk Tuesday in a conference call ahead of this weekend's game against the San Diego Chargers.
"I really honestly don't even have any idea about that rumor," McDaniels said. "I heard it. But I have nothing to add to it. My focus is solely on the Chargers and getting our team ready to go on offense. I'm trying to do the best that I can."
Information from ESPN.com's Mike Reiss and The Associated Press is included in this report.