With 306 seats in the House of Commons, David Cameron's Conservatives are still 20 short of a bare majority. Backing from the 57 Liberal Democrat legislators would give the Conservatives a comfortable cushion for passing legislation.
Simon Hughes, a senior Liberal Democrat legislator, said the talks among the party's legislators would continue through the weekend.
"There won't be a deal on the table because the talks have only just begun, but we will discuss where we want to go," Hughes said in a BBC radio interview.
"Everybody in Britain will expect us to be responsible. We know what the timetable is, it's between now and next Wednesday when Parliament comes back," Hughes said.
The two parties have important disagreements, including on whether to build a replacement for the nuclear-armed Trident submarine fleet and whether to reform the nation's voting system.
Conservative legislator Liam Fox said Saturday that such issues are less important than rebuilding Britain's recession-battered economy.
"It would seem to me very strange in an election that was dominated by the economy ... if the government of the U.K. was held to ransom over an issue that the voters did not see as their priority," Fox said in a BBC radio interview.
One possible deal is a coalition government, in which the Liberal Democrats would take a few Cabinet posts, or the Liberal Democrats could strike a bargain to support a minority Conservative government on key issues. Any deal would require backing from three-fourths of Lib Dem legislators and the same proportion of its executive.
For the time being, Prime Minister Gordon Brown remains in office despite his party's second-place finish in the election.
The election result offers the Liberal Democrats an opportunity to bargain for a change in a voting system that has given disproportionate influence to the two major parties, the Conservatives and Labour.
Liberal Democrats won 23 percent of all votes on Thursday, but only 9 percent of the seats in the House of Commons. The seat goes to the candidate with the highest vote in each of 650 districts. Liberal Democrats advocate a system common in continental Europe in which parties win seats in proportion to their share of the total vote -- a system that is much less likely to put one party in a dominating position.
"It would be inconceivable to me for the Liberal Democrats to sign up to anything that fell short of a guarantee of a referendum on reform of the voting system," Labour legislator Ben Bradshaw said on Saturday.
On Friday, Cameron offered the Liberal Democrats a committee of inquiry into the voting system -- which wouldn't necessarily lead to any change.
Brown has offered to legislate for a referendum on a change in the voting system.
Labour has previously encouraged talk of a change to proportional representation, but lost interest after winning a landslide victory in 1997 under Tony Blair.