David Davis keeps an ice pick in the corner of his sparse Commons office. But he denies that he might want to give David Cameron the Trotsky treatment and embed it in his rival’s head. Instead, he confides that it has sentimental value: “I go fell walking and that once stopped me falling over a cliff edge.”

He may need it again soon: for months Davis has been the odds-on favourite to take over from Michael Howard and become the next party leader. But that was before he was eclipsed at party conference by Cameron, the 39-year-old pretender. A poll for The Sunday Times last week found his popularity among Conservative members falling. Davis is once more teetering at the edge.

This week MPs will hold two ballots to decide which two candidates are to go forward for the 300,000-strong party membership to vote on. There are four runners left: Davis, Cameron, Kenneth Clarke and Liam Fox. A vote on Tuesday will eliminate one and a vote on Thursday will eliminate another.

Davis, the shadow home secretary, is still likely to poll most votes among MPs at Westminster. However, those MPs, many of whom had backed him because they presumed he would win, may well be thinking again. If he gets any fewer than the 66 that have officially declared for him, then he could be in serious trouble.

Since Davis’s poor performance at the Tory conference he has had to regroup. He has installed new lieutenants and gone back to trying to convince people why he is the answer to the Tory problem. Despite the momentum gathering around Cameron last week, Davis was in bullish mood as he defended his conference performance: “People criticised the speech but nobody said that there was no substance to it.”

He was having a dig at Cameron whose speech, although it made the blue rinses swoon, was what most consider “policy-lite”. However, it would seem that with the party in the doldrums, its members might welcome a bit of style over substance: “I had old ladies contacting my constituency office saying it has to be Cameron that we support,” an MP from a rival camp told me. “Cameron certainly did reach out.”

Davis needs to have a good last few days of campaigning and thinks that he can make headway by talking about his plans for tax cuts. He says there should have been more discussion about tax.

This weekend he is putting forward his plans for a low tax government. He believes that previous shadow Treasury teams have allowed themselves to be boxed into a corner by Gordon Brown: “Michael Portillo has admitted that he let him trap him in the static tax argument — that this tax cut needed this spending cut, which of course is not true.

“You need an agenda for growth. I will have my new growth rule. We will continue public expenditure but by less than the growth rate. The tax rate is the highest for 25 years.”

He makes the point by comparing the level of taxation here with that in Australia, where it is distinctly lower, noting that since 1997 the UK economy has grown by a quarter while in Australia it has increased by a third.

Davis has tomorrow’s hustings at the Commons to make one final push for support from his parliamentary colleagues. He still has the greatest number of MPs declared but he cannot afford to be complacent. His job is not to convince them to vote for him on Tuesday in the first ballot — he is almost certain to win that vote. He needs instead to make sure that he can keep their support in the second vote on Thursday.

The Fox camp, in particular, smells blood. “We know we are going to beat Ken Clarke in the first round and that leaves MPs with a choice,” one of Fox’s aides said. “There is now a feeling among many that Davis just cannot win among the membership, particularly if he is up against Cameron. That means they are going to have to decide whether to support the only other candidate that has the experience and credentials to be leader.”

The mathematics is always complex in these Tory battles. It is further complicated by MPs saying they are backing one candidate in public and in the secret ballot putting their cross elsewhere.