Hot on the Grantchester actor’s heels for the part are Richard Madden, Sam Heughan and Tom Hiddleston, all priced at 5-1.
Jack Lowden is at 6-1 to be the new Bond, while Tom Hardy is priced at 8-1.
There has also been speculation Bond bosses could introduce Idris Elba as the first black actor to play the iconic spy in a move away from the traditionally white, privileged stars who have held the title.
Norton, who studied at Cambridge University, also spoke out on how the public might react if he were cast as the next Bond.