Jones, 44, has a 50-27 record over six seasons as a head coach. He was the coach at Cincinnati for three seasons, going 9-3 overall and tying for first in the Big East Conference this year.
Tennessee fired Coach Derek Dooley in mid-November after the Volunteers were 4-7 in 2012. The Vols were a combined 15-21 since Dooley was hired before the 2010 season.
Tennessee officials were publicly turned down by two candidates. Oklahoma State's Mike Gundy said Wednesday he was remaining at the Big 12 school and Charlie Strong announced Thursday his intention to stay at Big East member Louisville.
Jones had his first head coaching position at Central Michigan, where his teams were 27-13 overall and 20-3 in the Mid-American Conference. He moved to Cincinnati before the 2010 season. The Bearcats went 4-8 in his first season but they were 10-3 in 2011.