The Prime Minister says Nick Clegg and David Cameron "should take as much time as they feel necessary" to agree on a coalition deal.
Watch election coverage live on Sky Player, including a statement from David Cameron.
Gordon Brown went on to say if the two leaders could not come to an understanding, he was willing to speak to Mr Clegg, whose position he "completely respects".
With the General Election resulting in a hung Parliament - the first since 1974 - party leaders are attempting to form a stable administration.
In the first foray into deal-making, the Prime Minister earlier said he had asked the civil service to provide support for talks between parties.
But arriving back in London from his Sheffield Hallam seat, Mr Clegg said: "I think it is now for the Conservative Party to prove that it is capable of seeking to govern in the national interest."
The Conservatives said Mr Cameron would announce his intentions on how to form a government "that is strong and stable with broad support" at 2.30pm today.
The Prime Minister, speaking outside Number 10, said it was his responsibility to make any government work for the national good.
"David Cameron and Nick Clegg should clearly be entitled to take as much time as they feel necessary," he said.
"For my part, I should make clear, I would be willing to see any of the party leaders.
"Should the discussions between David Cameron and Nick Clegg come to nothing, I would be prepared to discuss with Nick Clegg the areas where there may be some measure of agreement between our two parties."
Mr Brown also made it clear the business of government would continue - with Alistair Darling to meet with EU finance ministers tonight over the Greece's economic crisis.
During a dramatic night in British politics, the Conservative leader claimed Labour had lost its "mandate to govern".
But Business Secretary Peter Mandelson appealed for patience and indicated there may not be a resolution until Monday.
"[The Prime Minister] has to stay at his post, we have to wait for the results and that will become clear during the course of the day," he told Sky News
"If it is a hung parliament... then either there has to be a minority government that takes its chance, or a combination of other parties - in this case - most like the Labour Party and the Liberal Democrats, would have to see whether they could form some arrangement instead.
"We are not at that stage yet. We have to be patient and wait for the final results."
After campaign polls suggested the Liberal Democrats would experience a surge in voter support, the party failed to realise their electoral dream.
Mr Clegg conceded: "We simply haven't achieved what we hoped."
However, the Lib Dem leader remains a 'kingmaker' in any discussions of a potential coalition.
He indicated this morning one of the most important issues for his party was the "broken" electoral system that had delivered a hung parliament.
"It simply doesn't reflect the hopes and aspirations of the British people," Mr Clegg said.
"I will continue to argue not only for the greater fairness in British society, not only the greater responsibility in economic policy making, but also for the extensive real reforms that we need to fix our political system."
Elsewhere, the Scottish National Party have ruled out doing business with the Conservatives but said they would be open to talking with Labour.