A cross-party deal for a new press regulator underpinned by statute appeared closer as Labour said it was "confident" reopened talks had secured the basis of an agreement.
The Prime Minister began last-ditch efforts to find an accord on Sunday as he faced a likely Commons defeat on the issue on Monday with around 20 of his MPs set to back a rival package put together by an alliance of his Liberal Democrat coalition partners and the Opposition.
Deputy Labour leader Harriet Harman said an agreement had been reached, telling ITV's Daybreak: "I think we have got an agreement which protects the freedom of the press, that is incredibly important in a democracy, but also protects the rights of people not to have their lives turned upside down."
Ms Harman added: "In fact, the entirety of the Leveson report in terms of its key provisions - tough, independent self-regulation - has actually been agreed.
"What we are going to have is a system whereby it is a Royal Charter, which is quite a sort of complex and old-fashioned thing but that kind of more or less... has got legal basis. Then we are going to have a bit of statute, a bit of law, which says ministers can't tamper with it."
The Conservatives were represented at the meeting by Cabinet Office Minister Oliver Letwin - who has been the key figure for the party in months of Leveson talks - and they were also attended by Labour leader Ed Miliband, Deputy Prime Minister Nick Clegg and shadow culture secretary Ms Harman.
A Commons showdown was set up by Mr Cameron when he dramatically ended negotiations to find a way to implement the recommendations of the Leveson Inquiry into phone hacking. He had said using legislation would "cross the Rubicon" and endanger press freedom, but appeared to soften his stance over the weekend
While Number 10 sources said the statutory underpinning was "unnecessary and undesirable", Mr Cameron signalled that it was not "a big issue of principle".
The Prime Minister, who met Mr Clegg to discuss the issue on Sunday night, is not thought to have yet signed off on any agreement, however.
"You don't have to choose should you have a free press or should you protect people from abuse by the press. You can actually have both, and I think that is the agreement that we have reached and we will put it to the House of Commons this afternoon and I hope that we won't have a vote because I hope that everybody will be agreed."