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.

• The 10 Ukip targets seats that could make Nigel Farage the kingmaker

• The 7 constituencies that could leave the Lib Dems a laughing stock

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 10 seats the Greens could win

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.