David Cameron, the leader of Britain's Conservative opposition, has made a "big, open and comprehensive" offer to the Liberal Democrats in an attempt to form a majority government after the comments of Gordon Brown on such talks.
Speaking in London on Friday, Cameron said he hoped the two parties could reach an agreement quickly, admitting that the two parties do have substantial differences.
David Cameron approached the Lib Dems after the Tories won the most seats but finished 20 short of a majority.
His comments came after Gordon Brown, the prime minister, said he was willing to offer the Liberal Democrats electoral reform, one of their key policies, if they were to form a coalition with Labour.
The centre-right Conservative Party won most parliamentary seats but they need the support of other parties to form a stable government that can tackle the record budget deficit.
The negotiations between Liberal Democrat and Conservative parties could give the perennially third-ranking Lib Dems their first taste of power for decades.
But grassroot party members, more left-leaning than the Lib Dem leadership and who have the power to scupper any deal, said they were unhappy about the talks with the centre-right Conservatives.
"I will never consider voting for the Lib Dems again if a Conservative/Lib Dem pact is the outcome of this election," said one supporter writing on an Lib Dem activist website. "A Lib-Con coalition means nothing and will do nothing," wrote another.
Leader Nick Clegg must overcome scepticism among a significant number of his party, who fear that Britain's third largest party would be forced to sacrifice too many cherished policies for a deal.
Financial markets, already rattled by a debt crisis in Greece, want a new government to be formed quickly so it can set about reducing the deficit swiftly and decisively.
The pound, gilts and stock market all fell on Friday when it became clear the Conservatives would not have a parliamentary majority, despite beating the incumbent Labour Party soundly.
Conservative leader David Cameron appealed to the Liberal Democrats on Friday, saying he would consider some kind of formal agreement with them.
This could include a coalition, a rarity in Britain, but is more likely to involve a pact whereby the Lib Dems agree to support a Conservative-led minority government implementing an agreed legislative programme in return for concessions.
Senior members of both parties met on Friday night and further talks are expected over the next few days.
The most important hurdle is agreement on the pace of lowering the budget deficit. The Conservatives have pledged to start cutting it immediately but the Lib Dems warn this could harm Britain's recovery from a deep recession in 2008-2009.
Electoral Reform Debate
Electoral reform, along with immigration, Britain's role in the European Union and defence, are also likely to be stumbling blocks. The Lib Dems have long pushed for a change to the first-past-the-post voting system in favour of proportional representation.
Clegg has to persuade Lib Dem members of parliament it is worth making concessions, even though many party members are sceptical.
"I will never consider voting for the Lib Dems again if a Conservative/Lim Dem pact is the outcome of this election!" one supporter said on Liberal Democrat Voice, a web site for party activists.
Former Labour Deputy Prime Minister John Prescott said on Saturday the Conservatives, or Tories as they are also known, would not be able to offer proportional representation.
"We have introduced it in the Scotland and Wales regional governments so it seems to be on the horizon. But the Tories won't do it. We may," Prescott told Reuters on the sidelines of an environment conference in Beijing. "The best possibility for proportional representation is with the Labour Party."
If the Lib Dem/Conservative talks fail, a deal between Clegg's party and Labour was possible, but more complicated as the two parties combined would not have enough MPs to form a majority in the 650-seat House of Commons.
A Lib Dem-Labour deal would therefore have to involve other parties such as nationalist groups like Wales's Plaid Cymru and the Scottish Nationalist Party.
Labour leader Gordon Brown has said the Conservatives and Liberal Democrats had the right to try to form a government first even though he is allowed as sitting Prime Minister to have the first try under Britain's constitution.
His party has pledged to hold a referendum on electoral reform.