Allen, who was inducted into the Pro Football Hall of Fame last night, came up against White early in his career and was dominated by the former Philadelphia Eagles and Green Bay Packers player.
In his Hall of Fame speech, Allen revealed that he made a vow never to be outplayed in such a way again, resulting in him becoming one of the strongest players that the NFL has ever seen.
"I'd like to thank [Hall of Fame vice-president] Joe Horrigan, and the Hall of Fame staff for making this and helping us plan this weekend. At dinner, he brought up a story about a game I had against the great Reggie White," said Allen.
"I have never been dominated like that before in my life. So I got in that weight room and became the strongest man in the NFL, and I did it naturally all right.
"What's funny is once I benched 700 pounds, they tested me twice a week for the rest of my career."
Allen won one Super Bowl crown and was named in the NFL Team of the Decade for both the 1990s and 2000s during his illustrious career.