James May and Richard Hammond have been offered deals in the region of £1m a year to stay with the hit motoring show, according to The Times.
Such a deal would make the pair among the corporation’s highest earners.
It comes amid speculation the BBC will not directly replace Clarkson, but instead have a guest host on each show.
A BBC spokeswoman and representatives of the two men declined to comment, but bookmakers have slashed the odds on the pair returning.
William Hill spokesman Rupert Adams said: “Just a week ago we were offering 5-1 for them to return for another series of Top Gear but following news of the sums that have been offered to the pair, we have slashed the odds to 5-4.”
All three men’s contracts ran out earlier this year and the BBC refused to renew Clarkson’s after his much-publicised attack on producer Oisin Tymon.
Top Gear’s former executive producer — and old school friend of Clarkson — Andy Wilman has left the show, which sparked rumours he was joining up with all three men to make an alternative programme.
Several names have been put forward as possible replacements for Clarkson, including Guy Martin and Jodie Kidd, but another contender — Dermot O’Leary — has ruled himself out.