Lib Dem leader Nick Clegg has met Conservative leader David Cameron for talks over 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.
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.
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".
And he 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.
He added: "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 said: "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.
‘Brown should quit’
Mr Clegg said the Conservatives, as the biggest party, had the right to seek to form a government first.
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 Mr Clegg 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".
The Lib Dems have denied suggestions from a senior Lib Dem source of an angry phone conversation between Mr Brown and Mr Clegg. A Lib Dem spokeswoman said it was "perfectly amicable".
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.
This article is from the BBC News website. © British Broadcasting Corporation, The BBC is not responsible for the content of external internet sites.