Norton stars in the Sally Wainwright drama as rapist Tommy Lee Royce and has received much acclaim for his performance across the show's 18 episodes including a Bafta nomination for Best Supporting Actor.
Norton now sits as second favourite to replace Craig behind former Superman actor Henry Cavill, who nearly landed the part in 2006.
According to Ladbrokes, Norton is now 3/1 to become the new 007 though at 37, he may be considered too old to take on the role given producers have hinted that they wish to go younger with the character.
Norton has previously spoken about the potential of playing Bond as a diabetic (a condition the actor has). He said: "I’m a type one diabetic. How cool would it be to have a diabetic Bond? It would be wonderful for all those diabetic kids out there to think even a type one diabetic could be Bond!"
Norton also praised Craig's take on Ian Fleming's super spy: "I think Daniel Craig’s done an amazing job re-defining it for a modern era. And it’s this human role in a really wonderful way."
Happy Valley fans have also called on Norton to get the role with one tweeting: "Hope James Norton gets a crack at the Bond gig. Been superb in both #McMafia and #HappyValley."
Another viewer tweeted: "If James Norton doesn’t get Bond then GIVE HIM THE VILLAIN ROLE."
Broadcaster India Willoughby also wrote on social media: "James Norton to be the next James Bond."
Other contenders with the bookies to take over the role include Aaron Taylor-Johnson who was reported to have met with producers to discuss taking over the part. Tom Hardy and Bridgerton star Rege-Jean Page are also in the running according to bookmakers.
Longer shots to replace Craig include Gangs of London actor Sope Dirisu, former Game of Thrones star Richard Madden and Jack Lowden. Idris Elba has long been rumoured for the role but at 50, is considered too old.
Watch below: Is James Norton the next James Bond?