The two stars have been neck and neck for the last few weeks, with Norton having a slight lead at the bookies in the latter half of February.
But now Hardy has taken a strong lead over the McMafia actor to be the next James Bond.
Coral Bookmakers have the Dunkirk star at 3-1, while Norton is behind with 7-2.
Just behind them is Jack Huston at 6-1 and Idris Elba and Aidan Turner at 8-1.
Daniel Craig, 50, is set to shoot his fifth and final Bond film later this year.
James Bond 25 will be released in November 2019.