Some allies of Mr Davis suggested he could withdraw from the scheduled run-off among the Tories' 300,000 members if Mr Cameron wins a big majority among the party's 198 MPs. Under the party's rules, the MPs are due to choose a shortlist of two names, with the result of the members' ballot announced on 6 December.
If Mr Davis stood aside, Mr Cameron would be crowned party leader tomorrow without a contest - as Michael Howard was two years ago when the party dumped Iain Duncan Smith. The coronation would cap a spectacular rise by the 39-year-old shadow Education Secretary, who three weeks ago was an outsider in the Tory leadership stakes.
The Davis camp insisted that the shadow Home Secretary intends to fight on in a members' ballot. But one ally told The Independent: "If Cameron gets 60 per cent of the MPs, there is no way the members would overturn that. There wouldn't be any point in kicking lumps out of ourselves for two months, when we could be kicking the Government."
Mr Davis is locked in a tough fight with Liam Fox, the shadow Foreign Secretary, for second place behind Mr Cameron in today's vote. Although Mr Davis topped the poll in the first round on Tuesday, Mr Cameron's bandwagon continued to roll yesterday as he gained another 11 recruits, and he looks certain to come first today.
A spokesman for Mr Davis denied he would withdraw, saying: "He is fighting for every vote, confident of going into the next round, and he intends to take the fight to the country from Friday."
The Davis camp accused the Fox campaign of spreading black propaganda. One aide insisted: "They are trying all sorts of dirty tricks. There is no way that David is going to pull out. The party would feel cheated, and he would be pretty foolish to fall for that."
There was no sign of huge defections from Mr Davis to Dr Fox, who trailed the shadow Home Secretary by 20 votes in the first round.
Davis allies admitted they were down in the dumps after losing five votes on Tuesday. "There were lots of people running around like decapitated chickens, but the mood has stabilised," said one. "The important thing is, our support has not dissolved. We've lost one I know about, but there's been very little movement."
The Fox campaign was confident, however, that it was gaining enough support "below the radar" to push Mr Davis into third and win a place in a shootout with Mr Cameron. Fox aides said he would not stand aside to allow Mr Cameron to be acclaimed as party leader because he would have the momentum to beat him in the members' ballot.
Dr Fox said: "It would be unthinkable if our members were denied their say in the leadership election. Those who favour such an approach demonstrate an astonishing arrogance. I will do all I can to defend our members' participation in the ballot."
The shadow Foreign Secretary received a boost when he won the backing of Ann Widdecombe, the outspoken former home office minister, who supported Kenneth Clarke in the first round in which he was eliminated. But it was Mr Cameron who looked set to pick up most of the 38 MPs who backed Mr Clarke on Tuesday. Those who switched to him yesterday included Andrew Lansley, Stephen Dorrell, Adam Holloway, Sir George Young, Michael Jack, John Bercow, David Curry, Charles Hendry and John Horam.
Stewart Jackson, who backed Mr Davis in round one, changed horses to Mr Cameron. He said events had "moved on" since he declared his support for Mr Davis last month. "It is not personal, but I believe David Cameron has shown that he has the vision, charisma and the policy ideas to reach out to those groups who have stopped voting Conservative, such as women and professional people," he said.
Mr Cameron won a surprise recruit in David Heathcoat-Amory, a prominent Eurosceptic, who admitted he voted for Mr Fox on Tuesday in an attempt to knock out Mr Clarke. Another Clarke supporter, Quentin Davies, switched to Mr Davis.
The three contenders held a series of one-to-one meetings with wavering MPs and used BBC interviews last night to make a final pitch for votes.