The actor, who previously appeared in Happy Valley, has always been considered a contender, and has seen his odds shorten dramatically in recent weeks.
Now, it seems he’s the clear frontrunner.
Coral have him installed as the 1/2 favourite, now pulling comfortably ahead of Tom Hardy on 7/2.
“James Norton’s stock as an actor is increasing all the time and we feel it won’t be long before he is approached to replace Daniel Craig as the next 007,” said the company’s spokesperson John Hill.
Other names in the frame include Jack Huston on 4/1, Aidan Turner on 6/1, Tom Hiddleston on 12/1 and Idris Elba on 14/1.
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Punters can also get 5/1 odds that the next person to play James Bond will be female.
For now, Daniel Craig remains in the role for one more film, set to be released in 2019.
He finally confirmed speculation that he would reprise the part last summer; but added that it would be his final time.
Norton’s McMafia debuted on the BBC last night, and he admitted in a recent interview that he was ready for the Bond talk to kick up a gear.
“It’s really humbling and flattering, but to have my name next to the likes of Tom Hardy and Michael Fassbender is just mad,” he told Radio Times.
“If you’re thinking of putting a bet on me, keep your money in your pocket.”