The former Saturday Kitchen host is the 1/2 odds on favourite with William Hill to take over from the 50-year-old, after he quit the motoring series this week.
The 44-year-old, who left the cookery programme in March, is closely followed by motorcycle racer Guy Martin, 34, on 6/1.
"We have been swamped with bets for James Martin and I would be astonished if he does not get the nod," said William Hill spokesman Rupert Adams.
Meanwhile, Jenson Button, Rebecca Jackson and Philip Glenister are also in the frame with odds of 8/1.
Related articles
A BBC spokeswoman later told the Evening Standard: "We have no immediate plans to replace Chris Evans and do not expect to do so going forward.
"Series 24 starts filming in September."
James is yet to comment on the rumours and this morning ditched his car for something a little speedier, sharing a snap from a helicopter with the caption: "Up and away. See you in a bit!"
Express.co.uk has contacted a spokesperson for comment.