The well-known NASCAR racetrack has a capacity of 160,000, which means the game could set a new record for attendance in a team sport.
Tennessee spokesman Jimmy Stanton said he could not confirm the reports by ESPN reporter Joe Schad and CBS Sports reporter Bruce Feldman.
Bristol Motor Speedway has scheduled a press conference Monday morning to make a "major announcement."
On and off discussions about a game between the Vols and Hokies have existed for more than a decade.
UT previously said there were serious logistical issues that made playing a game difficult.
Tennessee coach Butch Jones was grand marshal of the Food City 500 at Bristol in February.
More on this story coming soon.
Evan Woodbery covers Tennessee football. Follow him on Twitter.