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Martin, the fan-favorite quarterback of the 1998 national championship team, has an offer from Jones to coach receivers, several websites reported.
Martin just finished his first season coaching receivers at Southern Cal. He would be a popular addition to the staff with UT fans.
Jones, who was hired Friday from Cincinnati, said having a Tennessee influence was an important component of his staff. He has apparently secured one link by retaining running backs coach Jay Graham from Derek Dooley's staff.
Paul Harris, a UT receiver commitment, also used his Twitter account to say Martin had an offer, although Harris may have been merely repeating Internet chatter.
Thigpen, 41, was hired Sunday, according to al.com.
Thigpen is from Virginia and was an All-Atlantic Coast Conference linebacker at North Carolina. He has coached for Urban Meyer at Bowling Green, Ron Turner at Illinois and returned to his alma mater to coach linebackers from 2005-08 before Chizik hired him at Auburn.
Willie Martinez, another Chizik assistant, was reported Saturday to be in consideration for a position on Jones' defensive staff.
If Thigpen and Martinez are on board, the defensive staff would appear set if Jones brings John Jancek and Steve Stripling from Cincinnati.
Jancek was defensive coordinator at Cincinnati and his name has been removed from the program's website. Previously he was on Georgia coach Mark Richt's staff from 2005-09, but previously was at Central Michigan when Jones was offensive coordinator.
Stripling coached the defensive line at Cincinnati and Central Michigan.
The offensive staff would have two openings if Graham is retained and Jones brings Mike Bajakian and Don Mahoney from Cincinnati.
Bajakian was offensive coordinator/quarterbacks coach and also worked for Jones at Central Michigan, helping develop Dan LeFevour.
Mahoney coached the offensive line at both Cincinnati and Central Michigan. He would be UT's third O-line coach in as many years for a group that returns four starters.
One opening would be receiver, where Martin would appear to be the favorite. If not Martin, then T.J. Weist would be a logical addition from the Cincinnati staff.
The other opening is tight ends. Jones has two options from Cincinnati.
Dave Johnson, a former Georgia aide, coached tight ends last year. Mark Elder previously coached tight ends but switched to safeties last year. Elder also coordinated special teams.
McKeefery, meanwhile, wrote about being fired on his personal website Sunday.
"It is also important to understand that this business is tough and gets harder to stay in," wrote McKeefery, who was earning $240,000 this season. "I recently spoke at the CSCCA National Convention and asked a question to the round table discussion I was leading "How many of you can name 5 Strength Coaches that have retired being strength coaches?". Maybe 5-10 people out of 300+ could. At 36 years old I know I will rebound from this position, but what if I was 46 or 56? It is important that we think about our mortality in this business and how we can make it better or what are some alternative to it."
McKeefery's departure means Tennessee will hire its sixth strength and conditioning coordinator since 2008.
Dave Lawson was assistant AD/Sports Performance at Cincinnati and previously worked for Jones at Central Michigan.
Darin Hinshaw, Dooley's receivers coach and recruiting coordinator in 2012, was linked Sunday by Internet reports to Kentucky, where new coach Mark Stoops is assembling a staff.