Final results on Friday gave the opposition Conservatives the lead with 306 seats in parliament, while Labour trailed second with 258 and the Liberal Democrats took 57.
One further seat is to be decided on May 27.
Despite gaining the largest share of the vote, the Conservatives are still 20 seats shy of the 326 needed to form a majority government. Clegg could bring 57 seats, more than making up for the shortfall.
If the two parties fail to find common ground, David Cameron, the Conservatives' leader, could go it alone, but a minority government could find it harder to pass laws.
'Open and comprehensive'
Cameron made a "big, open and comprehensive" offer to the Liberal Democrats in an attempt to form a majority government on Friday.
Speaking in London, he said he hoped the two parties could reach an agreement quickly, admitting that the two parties do have substantial differences.
The Conservative party's offer came after Gordon Brown, the prime minister, said he was willing to offer the Liberal Democrats electoral overhaul, one of their main policies, if they were to form a coalition with Labour.
But Al Jazeera's David Chater, reporting from London on Saturday, said Brown would be watching the talks carefully.
Final results -
Liberal Democrat: 57
"Gordon Brown is sitting in Number 10, probably for the next few days, waiting to see exactly how the negotiations go between the Liberal Democrats and the Tory party," he said.
Clegg had earlier said that the Conservatives, having won the largest number of seats, had "the first right to seek government".
The Conservative party's press office told Al Jazeera that "quiet negotiations" had begun with the Liberal Democrats on Friday.
Cameron said his party had priorities in common with the Liberal Democrats, which could provide "a strong basis for a strong government".
He cited the scrapping of ID cards, enhanced funding of disadvantaged schoolchildren with a "pupil premium", support for low carbon industries and avoiding the National Insurance rise.
But he has so far refused to soften his stance on the European Union, immigration or defence - areas where the Liberal Democrats hold counter views.
No olive branch
Vincent Moss, political editor of the Sunday Mirror, said Cameron has not offered the Liberal Democrats the olive branch they want on electoral reform.
However, he said Cameron is still in the "best position because he has the greatest number of seats".
Labour, which is more amenable to demands for electoral reform, would still be left a few seats short of a majority with the Liberal Democrats, meaning they would have to turn to Scottish and Welsh nationalists for further support.
|Brown said he is willing to offer the Liberal Democrats electoral reform [EPA]
Meanwhile, John Major, a former Conservative prime minister, has suggested that Cameron offer senior cabinet seats to the Liberal Democrats to tempt them into a coalition.
But speaking to Al Jazeera, Daniel Hamilton from ComRes, a polling organisation, said there was a "growing sense of anger among all political parties across the country that there is a feeling that some people, notably in some key marginal seats, appear to have been almost disenfranchised.
"I think whatever happens, whoever forms the government, there will be an urgent inquiry called by the leaders of all the three main political parties."
The failure of the election to produce a clear result was also likely to keep financial markets under pressure due to concerns about uncertainty over who will form the next government.