Nick Clegg, the Liberal Democrat leader, had earlier said that the Conservatives, having won the largest number of seats in Thursday's election, had "the first right to seek government".
Uncertainty remains over who will be Britain's next prime minister, after results showed no party had won enough seats to form a parliamentary majority.
Final results on Friday gave Conservatives the lead with 306 seats, while Labour trailed second with 258 and the Liberal Democrats took 57. One further seat is to be decided on May 27.
The Conservative party's press office told Al Jazeera that "quiet negotiations" had begun with the Liberal Democrats and that no statement or news conference was expected on Friday.
Cameron said his party had priorities in common with the Liberal Democrats, which could provide "a strong basis for a strong government".
Final results -
Liberal Democrat: 57
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.
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 added 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.
Al Jazeera's Tim Friend, reporting from Conservative party headquarters, said it was a "tricky moment" for the Liberal Democrats.
"They meet this weekend to discuss possible options. I guess it's possible they could open discussions simultaneously with both parties to see how far they go," he said.
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.
Earlier, Cameron said that the Labour party had "lost its mandate" to rule.
"I believe it is already clear that the Labour government has lost its mandate to govern our country," Cameron said.
|Brown said he is willing to offer the Liberal Democrats electoral reform [AFP]
While there was no clear winner, the election did throw up some interesting results, including the first ever seat for the Green party, represented by Caroline Lucas in Brighton Pavillion.
Two former Labour interior ministers, Jacqui Smith in Redditch and Charles Clarke in Norwich South, lost their seats.
The far-right British National Party failed to make headway in Barking, with Margaret Hodge, the sitting Labour MP, increasing her majority.
There were also angry scenes at some polling stations overnight when voters were turned away after large queues formed shortly before polls closed at 21:00 GMT.
Britain's electoral commission said on Friday that challenges to some national election results were likely as a result and that they had launched an investigation.
Problems with people being unable to vote were reported in areas including Milton Keynes in southern England, in Sheffield, the city where Nick Clegg holds a seat and Newcastle in northern England.
Police were called in London to one demonstration where 50 people refused to leave a polling centre after officials told them they were too late to vote.
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 is likely to keep financial markets under pressure due to concerns about uncertainty over who will form the next government.