He also was the Jags' interim coach for the final five games of the 2011 season.
Other candidates Fulmer reportedly is looking at are SMU coach Chad Morris, Washington coach Chris Petersen and defensive coordinators Jeremy Pruitt (Alabama) and Brent Venables (Clemson). He went on to coach at Nebraska, the Carolina Panthers, and Baylor before he coached at Florida State and Alabama.
All three coordinators are coming off of successful seasons.
Martin, 39, quarterbacked Tennessee to the 1998 national championship and played under Fulmer for the Vols.
Tucker is in his second season with the Bulldogs and now is helping prepare the team for its first appearance in the College Football Playoff.
Pruitt spent most of his early coaching career in Alabama at the high school and college level.
The coaching buzz aside, the attitude in the football building is "toxic" according to one assistant, with several employees knowing they will soon be fired once the coaching change takes place.
New Tennessee AD Phillip Fulmer is narrowing down the Vols' coaching search. Pruitt is now Alabama's DC.
If Martin does not land the Tennessee head coaching position - and he remains in the mix, although on the periphery - it's obvious there are plenty of people around the program who want to see him back at the school and positioned for the head coaching job in the future.
At a news conference for the College Football Hall of Fame on Tuesday, former Vols star Peyton Manning addressed the coaching search: "Obviously it's been a hard time, but I know what Tennessee is capable of being: It's what it was when I was in school there".