Lib Dem leader Nick Clegg has met Conservative leader David Cameron for talks about the possibility of their parties forming a new government.

A Conservative Party spokeswoman said the private 70-minute talks were "constructive and amicable".

The Tories won most election votes and MPs but are short of a majority.

In an e-mail message to supporters, Mr Cameron said he would not be "rushed into any agreement" but may be able to give "ground" in some areas.

Gordon Brown remains prime minister and has offered the Lib Dems talks if no deal is reached with the Conservatives.

A Lib Dem spokesman said Mr Clegg and Mr Brown spoke by telephone on Saturday night at the prime minister's request, describing the conversation as "amicable".

'Satisfy supporters'

BBC political editor Nick Robinson said the talks between Mr Cameron and Mr Clegg at Admiralty House in Westminster followed an earlier brief conversation at the VE day commemoration event in London.

Explaining how the two party leaders managed to meet without being spotted, our correspondent said Mr Clegg left the Work Foundation in central London at 1830 BST and aides said he was heading "home".

Meanwhile, Mr Cameron slipped away from Conservative HQ at about 1845 BST, unobserved by the media outside.

Our correspondent said it was the first opportunity the two men have had since the election to "look each other in the eye" and "judge whether the other one is being sincere" to see whether they can "actually do business together".

I hope we can sort things out as quickly as possible, for the good of the country. But we won't rush into any agreement David Cameron Hurdles facing Cameron and Clegg

He said: "They will have been working out whether there is any room there on which they can meet which would allow them to both satisfy their supporters and have some sort of stable coalition or arrangement."

And, referring to the Labour leader's offer of talks, our correspondent added: "There are very senior Liberal Democrats saying to Nick Clegg, 'Get out of these talks with the Tories and go talk to Gordon Brown'."

The Conservative and Lib Dem negotiation teams will meet again at 1100 BST on Sunday and there will be a meeting of Conservative MPs at 1800 BST on Monday, the BBC understands.

In Mr Cameron's e-mail message, he said he would stand firm on his pledges not "to give more powers to Brussels, be weak on immigration or put the country's defences at risk".

But he said areas of common ground with the Lib Dems included "the need for education reform, building a low-carbon economy, reforming our political system, decentralising power, protecting civil liberties and scrapping ID cards".

The Tory leader said, in the "national interest", the Conservative Party may be able to give ground in areas such as the Lib Dem manifesto plan to reduce taxes on the lowest paid.

'Fair votes now'

Mr Cameron said: "Of course, we hope to see a similarly constructive approach from the Liberal Democrats - not least on the urgent issue of tackling the deficit."

He added: "I hope we can sort things out as quickly as possible, for the good of the country. But we won't rush into any agreement."

There was no direct reference to the Lib Dem desire for a referendum on voting reform, although on Friday Mr Cameron offered an "all party committee of inquiry on political and electoral reform".

Earlier, Mr Clegg discussed the Tory power-sharing offer with his party, the leadership of which has "endorsed in full" his decision to talk to the Tories first.

The Lib Dem leader said the Conservatives, as the biggest party, had the right to seek to form a government first.

It is worth remembering that all the precedents - 1910, 1923, 1929 and 1974 - point towards a minority government and not a coalition The options for a hung parliament

On Saturday afternoon, during Mr Clegg's talks with senior party officials, an estimated 1,000 people gathered outside the meeting in favour of electoral reform, chanting "Fair votes now".

Mr Clegg left the talks briefly to accept their petition, and told them: "Reforming politics is one of the reasons I went into politics."

Meanwhile, Labour frontbencher Peter Hain said it was "clear" that the Lib Dem leader and Mr Brown had "a lot in common" on the need for electoral reform - Labour has offered a referendum on changing the voting system.

But Labour backbencher John Mann called for Mr Brown to step down as Labour leader before the party conference in September - arguing his position "rules out the credibility of a Lib/Lab pact".

Similarly, Labour MP and former sports minister Kate Hoey told BBC Radio 5 live she could not see how Mr Brown could "continue as prime minister in any kind of coalition" because "he wasn't elected originally" and had now "lost over 100 MPs".

'Desperate attempt'

Scotland's First Minister, SNP leader Alex Salmond, called on the Lib Dems to join a "progressive alliance" involving Labour, the SNP and Plaid Cymru.

However a Labour source dismissed that as "a desperate attempt by Alex Salmond to make himself look relevant after a terrible general election result".

The Tories secured 306 of the 649 constituencies contested on 6 May. It leaves the party just 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. The Conservatives got 36.1% of votes (up 3.8%), Labour 29.1% (down 6.2%) and the Lib Dems 23% (up 1%).

Meanwhile, a YouGov poll for the Sunday Times suggests more than two-thirds of people want Mr Brown to leave Downing Street immediately.

The poll of more than 1,400 voters found people think he should have admitted defeat on Friday, rather than hanging on in case the Conservatives cannot come to a deal with the Liberal Democrats.