The report said Allen would also be either president or vice president of football operations.
Allen, the son of Hall of Fame coach George Allen, spent the last nine years in the Oakland Raiders' front office. He had the title of senior assistant last season and was the team's salary cap manager and dealmaker with free agents.
The move to Tampa Bay reunites Allen with Gruden after the two worked together with the Raiders for four seasons.
Even with Allen as GM, Gruden reportedly will have more authority over personnel decisions and the draft than he had in the past. Allen is expected to handle contract negotiations and manage the salary cap.