Best known for playing rough-and-ready DCI Gene Hunt, Glenister has replaced Great British Bake Off host Sue Perkins as most likely to take on the role, according to bookies.
Odds have been slashed from 10-1 to 2-1 after reports that the BBC is considering hiring the actor-turned-presenter.
The BBC declined to comment on the speculation.
Glenister is a vocal car enthusiast and is about to present the second series of For the Love of Cars on Channel Four.
David Williams of Ladbrokes said: “All of a sudden Glenister is the hottest name around.
“We hadn’t seen a penny for him before the weekend but in next to no time he’s the new favourite. Punters seem convinced that something is afoot.”
Simon Clare, spokesman for Coral, said that it had slashed odds from 10/1 to 2/1 for Glenister.
“We have seen a host of celebrities gambled on in recent weeks to take over from Clarkson and it is rapidly becoming one of the biggest novelty betting events ever”, he said.
Behind Glenister, Perkins is the next favourite at 4-1 followed by Dermot O’Leary, who has recently quit X Factor, at 5-1.
Other contenders include Jodie Kidd, Steve Coogan, John Bishop, Johnny Vaughan and Stephen Fry.
Clarkson was the subject of an internal BBC investigation after he attacked Top Gear producer Oisin Tymon, splitting his lip and verbally abusing him in a 30-second assault on 4 March.
He was suspended by the broadcaster on 10 March before it announced his contract would not be renewed.
The star received widespread public support, including from his friend Prime Minister David Cameron.