There is no suggestion of a coup against Mr Cameron, with those involved describing the talks as “sensible contingency planning” in case the party needs a new leader quickly. Mr Cameron himself has suggested that he would have failed as leader if he is unable to form majority government after the election.
The Telegraph has spoken to senior Conservatives from different parts of the party - ministers, backbenchers and party officials. Several suggested that Mr Cameron could step down even if the Tories win more seats than Labour at the election.
One option being examined is for Mr Cameron to remain Prime Minister for a short period while the Conservative Party arranges a “coronation” for Boris Johnson as leader, who would then take the premiership and try to assemble a Tory-led minority government.
Another scenario would see the Conservatives decline the chance to try to form a government if the numbers were not in their favour, instead allowing Labour the chance to do a deal with the Scottish National Party in the hope that the new government would quickly collapse.
Polling analysis suggests that the Conservatives could well be the largest party in parliament, beating Labour but still falling short of a Commons majority of 323 seats.
Boris Johnson at a children's play centre in Surbiton
However, the party’s ability to form a government could be limited by its potential partners.
Of the smaller parties, only the Liberal Democrats and the Democratic Unionists have suggested they could be willing to work with the Tories to support a minority government. Those two parties are expected to have fewer than 40 seats between them on current polling.
Some senior Tories are sceptical of whether the Lib Dems would actually back a Tory government, given the political harm the party suffered from its last coalition deal with the Conservatives.
Labour, by contrast, could work with the Lib Dems and the SNP, who between them could have 80 MPs, likely to be enough to give a minority Labour government a working majority in the Commons.
Mr Cameron took the Tories into power in 2010 despite failing to win a majority.
As the incumbent Prime Minister, Mr Cameron would have the first attempt to form a government in a hung parliament after this election.
But one senior Tory figure suggested that, if the Tories had most seats but not enough allies for a clear majority, the party might decide to remove him rather than let him make the attempt.
Instead of a formal vote of no confidence, a delegation of senior MPs would tell Mr Cameron to resign, under this plan.
MPs believe the PM would not resist. They noted that in a BBC interview last week, he appeared to accept that if he did not win a majority he would have failed as a leader.
“If the numbers aren’t there, there would a very strong argument for saying ‘You’ve now failed to win two elections in a row – time’s up”.” said the senior Tory.
Some Tories believe that a Labour government backed by the SNP would very unstable and potentially collapse within months.
Under current laws, that would not automatically mean a new election, but might give the Tories another chance to try to form a government.
Some Tories want to make sure the party is ready with a new leader - Mr Johnson - as soon as possible after the election.
The last Conservative leadership election took two months as MPs held several ballots to select a shortlist of two that was then put to party members in a postal vote.
One minister said that Mr Johnson now has a “very well organised team in place” to make him leader quickly after the election.
The team is said to include at least one Cabinet minister and several well-placed MPs, as well as leading Conservative thinkers and pollsters.
“They’re ready for all the options," said the minister. “There is talk of Boris being appointed by acclamation.”
One option would see Mr Cameron resign as leader quickly after the election but stay on until Mr Johnson was "acclaimed" Tory leader without a formal election among the full party membership.
That might mean a quick vote among MPs, or even a return to the Conservative tradition of a "magic circle" of grandees picking a leader, a practice that continued until 1963.
Boris Johnson and the Tory leadership question
April 22 2015
This week Boris said it would be "wonderful" to be prime minister one day. He said: "It is at least five years away which is an aeon in British politics, by which time whatever my personal ambitions may be there will be thrusting young men and women who will be overtaking me and who knows, it will all be different. In the dim, distant future, obviously it would be a wonderful thing to be thought to be in a position to be considered for such an honour but I think it highly unlikely."
September 15 2014
On announcing his campaign to retain the seat of Uxbridge and South Ruislip for the Conservatives, Boris sought to play down suggestions it marked the latest stage of a mission to succeed David Cameron as Tory leader. He said: "No, this is the first stage in the campaign to retain the seat of Uxbridge and South Ruislip for the Conservatives. This is Act One, Scene One of a very long process. There's a lot of digging in to be done."
August 6 2014
As speculation began to build when Boris announced he planned to run for parliament, he was forced to refute claims he was building up to a bid for the Conservative leadership. He said "I think it's highly unlikely that that will happen because there's no vacancy. I think David Cameron has been a brilliant prime minister. When David Cameron finally steps down, in 2030, or whenever, it may be that there's a vacancy, but it will probably be filled by a person who's a teenager now."
October 1 2013
Boris encouraged speculation about his plans for the Tory leadership when - hours after David Cameron said he would support him in returning to the House of Commons as an MP - Boris referred to Alain Juppé, the former French prime minister and said: “He told me that he was now the mayor of Bordeaux – I think he may have been mayor of Bordeaux when he was Prime Minister. It’s the kind of thing they do in France.”
March 25 2013
Asked in an interview for a BBC documentary if he would like to be prime minister one day, Boris said: "Well I would like to be the lead singer of an international rock group. That was my aim. Or a guitarist. I would love to have been a world famous painter or a composer. There are many many things I have done or would like to have been able to do. Obviously, if the ball came loose from the back of a scrum - which it won't - it would be a great, great thing to have a crack at. But it's not going to happen."
March 22 2013
Speaking to schoolchildren at south London’s Norwood School, Boris said: "If, like the Roman leader Cincinnatus, I were to be called from my plough to serve in that office, I wouldn't, of course, say no." He went on to repeat his familiar denial, saying that the chances of his actually becoming prime minister were "about as good as my being reincarnated as an olive". However, he added: "If people genuinely wanted me, of course I would want to do it."
June 2 2012
While being quizzed by crowds at the Hay Festival back in 2012, Boris said: “As I never tire of saying, my chances of becoming prime minister are only slightly better than being decapitated by a frisbee, blinded by a champagne cork, locked in a fridge or being reincarnated as an olive.”
November 25 2009
"Were I to be pulled like Cincinnatus from my plough, then obviously it would be an absolute privilege to serve."
October 5 2005
Speaking about the Conservative leadership contest in a conference diary he wrote for The Independent back in 2005, he said: "I'm backing David Cameron's campaign out of pure, cynical self-interest."
June 17 2004
When asked by a reader in The Independent to admit that he wanted to become prime minister, Boris said: "My chances of being PM are about as good as the chances of finding Elvis on Mars, or my being reincarnated as an olive."
Boris is quoted by his sister Rachel Johnson as having had ambitions to be "World King" as a child. She said: “As Boris was growing up whenever anyone asked him what he wanted to be, he would answer: ‘World King’.”