The process of forming a new government may be resolved in the next 24 hours, according to a senior Liberal Democrat source who said it was "crunch time".
The Lib Dems have held talks with the Tories and Labour in a bid to form an alliance which could run the country.
Gordon Brown, whose presence in Downing Street was seen as harming Labour’s chance of a deal, has announced he will step down as party leader by September.
No party won a Commons overall majority at Thursday’s general election.
Labour and the Tories are both trying to woo the Lib Dems with promises on electoral reform as the battle form a new government reaches its critical phase.
A meeting of Lib Dem MPs continued beyond midnight and ended with no firm decisions taken, the BBC understands.
Following Mr Brown’s announcement that he was standing down as Labour leader, the party’s deputy leader Harriet Harman became the first senior figure to say she had no plans to stand in a leadership contest.
The Tories, who won the most seats and votes in the election, reacted to Mr Brown’s decision by making a "final offer" to the Lib Dems of a referendum on changing the voting method to the Alternative Vote (AV) system.
Labour are offering to put the AV system into law and then hold a referendum asking voters to approve it.
BBC political editor Nick Robinson said Mr Brown’s resignation was an audacious bid to keep Labour in power – and the prime minister himself in power for a limited period – and that Tory MPs would be furious.
In his statement, Mr Brown said Britain had a "parliamentary and not presidential system" and said there was a "progressive majority" of voters.
He said if the national interest could be best served by a coalition between the Lib Dems and Labour he would "discharge that duty to form that government".
Mr Brown said no party had won an overall majority in the UK general election and, as Labour leader, he had to accept that as a judgement on him, before adding that he hoped a new leader would be in place in time for the Labour Party conference in September.
He has urged potential candidates, such as Foreign Secretary David Miliband and Schools Secretary Ed Balls, not to launch their campaigns yet.
Mr Brown said Lib Dem leader Nick Clegg had requested formal negotiations with Labour and it was "sensible and in the national interest" to respond positively to the request.
It emerged earlier that the Lib Dem negotiating team, who have held days of talks with the Conservatives, had also met senior Labour figures in private.
But it was understood that one of the stumbling blocks to any Labour-Lib Dem deal was Mr Brown himself.
Mr Clegg said he was "very grateful to David Cameron and his negotiation team" and they had had "very constructive talks" and made a "great deal of progress".
But he said they had not "reached a comprehensive partnership agreement for a full Parliament" so far and it was the "responsible thing to do" to open negotiations with the Labour Party on the same basis, while continuing talks with the Tories.
"Gordon Brown has taken a difficult personal decision in the national interest," he said.
"And I think without prejudice to the talks that will now happen between Labour and the Liberal Democrats, Gordon Brown’s decision is an important element which could help ensure a smooth transition to the stable government that everyone deserves."
The Lib Dems have long campaigned for a change to the voting system – something which the Conservatives have strongly opposed.
But speaking after a meeting of Conservative MPs, following Mr Brown’s statement, shadow foreign secretary William Hague said they were prepared to "go the extra mile" on electoral reform – and offer a referendum on switching to AV in return for a coalition government.
He said the Lib Dems had to choose whether to back them or a government that would not be stable – because it would have to rely on the votes of other minor parties – and would have an "unelected prime minister" for the second time in a row.
He also said the Labour offer was for a switch to the AV system, without a referendum, which he believed was undemocratic. The BBC understands, from Lib Dem sources, that the Labour offer is legislation to introduce AV, followed by a referendum on proportional representation.
Under AV no candidate is elected without at least 50% of the vote, after second preferences are taken into account, but it is not considered full proportional representation.
Meanwhile, BBC political correspondent Iain Watson said he understood some cabinet members were sceptical about the idea of a "progressive alliance" with the Lib Dems and were concerned it would look bad.
Labour backbencher Graham Stringer said he did not believe a coalition with the Lib Dems would work and could damage the party: "I don’t think it makes sense in the arithmetic – the numbers don’t add up."
The Tories secured 306 of the 649 constituencies contested on 6 May. It leaves the party short of the 326 MPs needed for an outright majority, with the Thirsk and Malton seat – where the election was postponed after the death of a candidate – still to vote.
Labour finished with 258 MPs, down 91, the Lib Dems 57, down five, and other parties 28.
If Labour and the Lib Dems joined forces, they would still not have an overall majority.
With the support of the Northern Irish SDLP, one Alliance MP, and nationalists from Scotland and Wales they would reach 328, rising to 338 if the DUP, the independent unionist and the new Green MP joined them.
Coalition scenarios
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