Although no talks have taken place, supporters of the two candidates for the first-round ballot on 10 June told The Independent yesterday that an alliance would help them both if the result of the first round was inconclusive.
The former Chancellor of the Exchequer is expected to come top of the ballot in the six-strong field that is expected next Tuesday, but he could find it difficult to win without the help of the centre-ground Mr Hague.
While no candidate can deliver his support for another, second-round candidate, en bloc, an endorsement could mobilise solid support for a leading contender.
It is increasingly being recognised at Westminster that at 36, Mr Hague could well be too young to lead old-hands like Mr Clarke, Michael Howard, Peter Lilley and John Redwood in a united shadow cabinet.
But if he was assured of a leading role in a Clarke team, possibly as deputy leader with responsibility for party organisation, he could prepare himself for an eventual succession to Mr Clarke, who is 57 next month.
Senior sources in the Clarke and Hague camps said yesterday that there could be no talks until after the results of the first round had been declared on Tuesday evening. But quick moves might be required in advance of the second-round ballot one week later, on 17 June.
There were strong rumours in Westminster yesterday - no doubt inspired by Clarke supporters - that if the party leadership was taken by a right- winger there could be a significant number of defections from the parliamentary party.
On the right-wing, it was said again yesterday by a leading Howard supporter that his man was running in second-place to Mr Cklarke. But it was also conceded that if he was beaten by Mr Lilley, then he would be left with no choice but to move behind him as the leading challenger to Mr Clarke.
However, the right-wing is by no means united. One of Mr Redwood's leading supporters told the Independent that Mr Lilley and Mr Hague had "no conviction, no passion, no guts and no brains."
He was particularly scathing about the "Mandelsonian hollowness" of Mr Hague, but his comments on Mr Lilley went into the regions of libel.
If that loathing is common in the Redwood camp, it is possible that Mr Clarke could pick up support from that surprising quarter in the second and third rounds of the contest, although the same, well-informed source thought it rather more likely that the man who finally fought Mr Clarke would emerge as eventual winner.
Yesterday, with just a week to go, John Major said he would not give public support to any of the six runners, Mr Hague announced that he had won the backing of the former Cabinet minister Lord Parkinson, while Mr Redwood set out his plans on public spending.
Mr Major defended his decision to step down immediately after the general election as leader rather than waiting until the party had had time to consider its future.
"I think the sooner we can prepare, the sooner we have a new leader and the sooner we can then turn all our attentions to the shortcomings of the new Government where we perceive there to be shortcomings, the better," he said.
Mr Redwood said: "We must both be the party of better schools and hospitals and the party of lower taxation.
"We can do it if we recommend sensible changes in the priorities in the budget."
He called for spending cuts of pounds 3bn to pay for extra money for schools, the National Health Service and help for small businesses.
Savings could be made by scrapping plans to ban handguns, thus cutting compensation costs, by abolishing Urban Development Corporations, but ending regional government offices and freezing civil service recruitment, he said.