The Liberal Democrat leader, Nick Clegg, has challenged the Conservative Party, which has the largest share of the public vote, to come up with a proposal "to govern in the national interest".
The call came this morning after it became clear that no party will have an overall majority in Westminster after the 6 May General Election, with 618 out of 650 seats declared.
The Tory leader, David Cameron, has said that he will make an announcement about his intentions at 2.30pm this afternoon.
As speculation intensified in the wake of a hung parliament, Downing Street told the BBC that Labour was also interested in putting forward a proposal.
Mr Clegg, whose party has won one per cent more votes than in 2005, but has claimed 10 per cent fewer seats (as a result of the idiosyncracies of the first-past-the-post voting system), takes the view that the party with the largest number of votes and seats has an obligation to try to find a way of governing with stability and consensus.
But according to tradition, the monarch will also accept proposals for a governing arrangement, involving other parties, from an outgoing Prime Minister - even one with fewer votes
Meanwhile, the Conservatives have achieved a massive 44 per cent increase in their number of parliamentary seats on the back of only a four per cent increase in the popular vote.
Political reformers gathered around the 'Take Parliament back' campaign (http://www.takebackparliament.com/hope ) say that this is indicative of an unfair and failed system - one that Mr Clegg has also called "broken".
One of the supporters of the initiative, the commentator and columnist George Monbiot, wrote this morning in the Guardian newspaper: "With no clear winner, the people's verdict is a plague on the old politics. Now is the time for real reform."
He continued: "The UK's locked-down politics has suddenly been flung wide open. For the first time in living memory, we have a chance to smash our antediluvian system. If we can seize the opportunity a hung parliament offers, to deliver proportional representation and party-funding reform, we will change politics in the UK for ever."
"Radical constitutional reform will happen only if we demand it, so loudly and so doggedly that parliament and government, whatever their composition might be, can no longer fend us off.
"The fight starts tomorrow, with rallies in London, Glasgow, Manchester, Bristol, Middlesborough, Oxford and possibly other cities. It is being coordinated by the kind of wide-ranging coalition we've needed for years, as almost all the major reform campaigns – Power2010, Make Votes Count, Unlock Democracy, the Electoral Reform Society, Ekklesia, Compass, Hang 'em, Vote for a Change and others – have ... come together."