Newspaper editors will hold crunch talks today after David Cameron warned that "the clock is ticking" to avoid regulation backed by law.
The Prime Minister has told the press it must act fast to convince politicians and the public that Lord Justice Leveson's call for statutory underpinning is not necessary.
The push for an agreement comes as Labour, the Liberal Democrats and victims of media abuses continue to demand full implementation of the judge's recommendations.
Cameron's own party is also seriously split on the need for legislation.
Speaking after meeting editors in Downing Street yesterday, the premier insisted any new regulator had to meet the report's requirements.
"That means million-pound fines, proper investigation of complaints, prominent apologies, a tough independent regulatory system," he said. "And they know, because I told them, the clock is ticking for this to be sorted out."
Culture Secretary Maria Miller said she expected the industry to come up with a process and timetable by tomorrow – amid suggestions that detailed proposals could be ready within a fortnight.
Lord Black of Brentwood, the Telegraph Media Group executive director who has been helping draw up reform plans, said it was crucial for the press to remain "united".
Speaking at Press Gazette's British Journalism Awards last night, he said. "To meet tough deadlines, which the Government is rightly imposing on us, it's going to take a great deal of determination, energy and commitment from the whole industry…
"One thing is going to be absolutely vital in the days ahead and that is that the industry must remain united as we seek to implement the Leveson report and put in place a new tough system of regulation which is fit for the future – and which above all else will make the need for statute irrelevant."
Independent editor Chris Blackhurst said there was "movement" across Fleet Street towards an agreement on making the new body fully independent of serving editors.
There was unanimity that it would need the teeth proposed by Leveson, he said, such as the power to levy £1 million fines, to launch investigations and direct the size and prominence of apologies.
And the industry recognised the need to return the "favour" done by Cameron in resisting statutory regulation, he told Sky News.
Lloyd Embley, editor-in-chief of the Daily Mirror, Sunday Mirror and The People, said there was "a firm belief that papers can deliver Leveson principles far more quickly without legislation".
That would be "better for public and free speech", he posted on Twitter.
Daily Telegraph editor Tony Gallagher likened the meeting to "the summoning of the Five Families in The Godfather".
Officials at Miller's Culture Department are drawing up a draft Bill to enact the Leveson recommendations in full.
But she has indicated that she expects it to confirm concerns about the complexity and potential negative impact on press freedom.
Labour is drawing up its own draft legislation to demonstrate that Leveson's recommendations could be implemented without the difficulties feared by ministers.
The opposition's Bill is expected to be completed before Christmas, paving the way for Labour to force a Commons vote by the end of January if no agreement is reached in cross-party talks. The result would not be binding on the Government, but a defeat could be hugely damaging for the Prime Minister.
As the editors met at Number 10, Labour leader Ed Miliband hosted talks in his office with representatives of the Hacked Off campaign group, including phone hacking victims.
"What I heard from the victims of sections of the press is that they are pleased there are good intentions from the editors but they want more than good intentions," he said.
"They want the force of law to make sure those good intentions are turned into reality."
He said he did not doubt the goodwill of the editors to reform, "but we have no guarantee that that goodwill and those good words are actually going to mean something on an ongoing basis".
Deputy Prime Minister Nick Clegg, who is opposed to Cameron's stance, said he hoped "gridlock" could be avoided in the continuing cross-party talks.
Speaking on a visit to a school alongside Mr Cameron, he said: "We owe it to the victims to be able to say to them, 'Look, we've got a solution and this won't happen again'.
"That's what I'm determined to do and that's why I'm entering into cross-party talks in good faith to get a solution that works for everybody."