Labour has rejected the idea of a “government of national unity” – headed by a figure like Ken Clarke or Margaret Beckett – to prevent a no-deal Brexit.
Shadow Chancellor John McDonnell said any interim government – formed after the removal of Boris Johnson – must be headed by Jeremy Corbyn.
He said Labour was “unlikely” to table a no confidence vote in the government until after 17 October’s EU summit.
Labour is talking to other opposition parties about toppling the government.
But there is deadlock over how best to to do this, with the SNP pushing for a no confidence vote as soon as possible.
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The Lib Dems have, meanwhile, said they will not back an interim government headed by Mr Corbyn, with leader Jo Swinson saying the Labour leader had to make clear who else he would support to temporarily take charge.
She has said Mr Corbyn “simply does not have the numbers” to command a majority in the Commons, referencing the 21 MPs expelled from the Conservative Party and the five members of the Independent Group for Change.
Talks between opposition leaders at Westminster on how to best prevent a no-deal Brexit are expected to continue on Wednesday, convened by Mr Corbyn.
The caretaker PM would have to be “a figure who is well respected and above the everyday party politics,” Ms Swinsons told the BBC.
Veteran Europhile and former Tory chancellor Ken Clarke and Labour’s Margaret Beckett, who was caretaker leader of her own party in the early 1990s following the death of John Smith, have both been suggested as possible caretakers.
But asked whether an interim government could be led by anyone other than Mr Corbyn, the official leader of the opposition, Mr McDonnell said: “No, the rules are the rules.”
He said Ms Swinson might change her mind on the issue, adding: “I’m a great believer in the powers of conversion.”
In response, a spokesman for Ms Swinson said she was “a great believer in the power of mathematics” and the sums did not add up for the Labour leader.
The opposition parties say they are united on their desire to prevent Mr Johnson from taking the UK out of the EU on 31 October without a deal.
Mr Johnson is urging them to vote for a general election but they say they will not do this until a no-deal Brexit has been ruled out – and they do not trust Mr Johnson to obey the law by asking for an extension at October’s EU summit.
The PM insists a deal is still possible with the EU and the government will set out fresh proposals in the next few days.
Mr McDonnell said the 21 former Tory rebels would “obviously” want to see what Mr Johnson gets at the EU summit before deciding whether to back a no confidence motion.
He said Labour was seeking a meeting with Cabinet Secretary Sir Mark Sedwill to try to establish what would happen if the government was defeated but Mr Johnson still tried to “squat” in No 10 until after Britain was out of the EU.
“We are suggesting that we meet with the Cabinet Secretary to just, at least, get the ground rules sorted,” he told reporters at Westminster.
SNP sources have expressed frustration that the cross-party talks on the next Brexit steps are becoming little more than “tea and biscuits” meetings.
A request from opposition party whips to have a debate over the release of the government’s no-deal Brexit planning papers was rejected by the Speaker John Bercow on Monday.
The SNP’s leader at Westminster, Ian Blackford, said opposition MPs who supported the law requiring the PM to seek a Brexit delay should have the “courage of their convictions” and back a no-confidence vote to remove the “toxic” Mr Johnson from Downing Street.
“That would be real leadership, if the opposition parties were prepared to do that,” he told BBC2’s Politics Live programme.
“We’ve got a prime minister that may be prepared to break the law and crash us out at the end of October,” he added.
“If we want to guarantee that we’re staying in Europe at the end of October, if we want to stop this prime minister, then we have to stand up and be counted – and that means a motion of no confidence”.
Mr Blackford also rejected suggestions an interim government could hold a further Brexit referendum before an election, adding it would be “extremely challenging”.
He said this would require any such administration to be in place for “at least six months”, but there was not “anything like a majority” in Parliament for such an idea.