Conservative leader David Cameron says it is "decision time for the Liberal Democrats" over which party they will back to form a government.
Mr Cameron, whose party won the most seats and votes but were short of a majority, said he hoped they would make the right decision.
It follows days of talks between the Conservatives and Liberal Democrats, after the UK election resulted in a hung parliament.
On Monday the Liberal Democrats formally opened talks with Labour about a rival deal.
The announcement about the Lib Dem/Labour talks came from Gordon Brown in the same Downing Street statement in which he said he would step down as Labour leader by September.
Labour and the Conservatives are both trying to woo the Liberal Democrats with promises on electoral reform as the battle to form a new government reaches its critical phase.
Labour say if the Liberal Democrats back them they will put the Alternative Vote system into law and then hold a referendum asking voters if they want a proportional representation voting system - a key issue for the Lib Dems.
The Conservatives upped their offer on Monday evening to the Liberal Democrats to a promise of a referendum on changing the voting system to the Alternative Vote system.
Speaking on Tuesday morning Mr Cameron said his overriding concern since Friday was for "strong, stable government that is in the national interest" and his party had made a "very reasonable" offer to the Lib Dems to deliver it.
He said his MPs had put aside party interest in favour of the national interest - after they approved a referendum on the voting system, a reform the Conservatives have always opposed.
Mr Cameron said: "It's now, I believe, decision time, decision time for the Liberal Democrats and I hope they will make the right decision to give this country the strong, stable government that it badly needs and badly needs quickly."
A meeting of Lib Dem MPs continued beyond midnight and ended with no firm decisions taken, the BBC understands.
Labour's ruling national executive committee is set to meet on Tuesday to discuss the prospect of a coalition and the time it will take to replace Mr Brown.
The Conservatives secured 306 of the 649 constituencies contested on 6 May. It leaves the party 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 Liberal Democrats 57, down five, and other parties 28.
If Labour and the Liberal Democrats joined forces, they would still not have an overall majority.
Here at Greengates Builders Merchants Accrington, Lancashire we are hoping a decision will soon be made about the government and then hopefully the country will be able to move onwards and upwards.