Labour and the Liberal Democrats both support statutory underpinning, but the prime minister and the majority of Conservative MPs are against it.
Labour and the Liberal Democrats both support statutory underpinning, but the prime minister, many Conservative MPs and parts of the press are against it.
In Scotland, MSPs are also debating press regulation and whether to take a UK-wide approach or create a separate system.
The Downing Street meeting was organised by the culture secretary, Maria Miller. After it had concluded, she said the industry had "responded positively" to the challenge of establishing a new regulator.
Tuesday's meeting was organised by Culture Secretary Maria Miller. After it had concluded, she said the industry had "responded positively" to the challenge of establishing a new regulator.
Asked how quickly she expected progress on a plan, she said: "The industry will be setting that out in the next two days."
Mrs Miller added: "The challenge has been thrown down to them. They have responded positively and it is now for them to go away and develop those plans.
In a statement, the Department of Culture added: "There is still a huge amount of work for the industry to do, and it must not drag its feet.
"The industry will be setting that out in the next two days. We are now in the middle of a process and I really think we need to make sure that process is completed."
"She is clear that, while she has very grave concerns about legislation, if the industry fails to deliver then it will be the only option left."
The chairman of the existing Press Complaints Commission, Lord Hunt, who also attended the Number 10 talks, has been leading the industry's attempts to draw up a replacement for his own body.
He claims 120 publishers, representing 2,000 editors, are supporting his plan and believes legally-enforceable membership contracts could be used to avoid the need for statutory backing.
But the department's statement said the editors at the meeting had been reminded that Lord Justice Leveson had found Lord Hunt's model "unconvincing".
'No beer'
'No beer'
Among editors attending were The Sun's Dominic Mohan, The Guardian's Alan Rusbridger, Geordie Greig from the Mail on Sunday, the Daily Telegraph's Tony Gallagher, James Harding from The Times, Sarah Sands from the Evening Standard, Lloyd Embley from the Daily Mirror, Sunday Mirror and The People and The Spectator magazine's Fraser Nelson.
Among editors attending were The Sun's Dominic Mohan, The Guardian's Alan Rusbridger, the Daily Telegraph's Tony Gallagher, James Harding from The Times, Lloyd Embley from the Daily Mirror, Sunday Mirror and The People and The Spectator magazine's Fraser Nelson.
Emerging from the meeting, Mr Gallagher tweeted: "19 editors and industry reps, 9 mandarins, 3 ministers and 1 PM. We got coffee and still tap water. No beer and sandwiches."
Mr Gallagher tweeted: "19 editors and industry reps, 9 mandarins, 3 ministers and 1 PM. We got coffee and still tap water. No beer and sandwiches."
He also added that the meeting "felt like the summoning of the Five Families" from the film The Godfather.
Spectator editor href="https://twitter.com/frasernelson" >Mr Nelson tweeted soon after the meeting: "Distance between Leveson's proposed regulation and what press is prepared to sign up to v small. Statute should not be needed to close gap."
href="https://twitter.com/LWEmbley66" >Mr Embley said there was "a firm belief that papers can deliver Leveson principles far more quickly without legislation. Better for public and free speech".
Chris Blackhurst, editor of the Independent newspaper, said there was "broad agreement" in many areas, but "one sticking point" concerned the independence of the watchdog.
After leaving the meeting, Mr Cameron said the industry had to take action and added: "They have got to do it in a way that absolutely meets the requirement of Lord Justice Leveson's report.
Mr Cameron said the industry had to take action and added: "They have got to do it in a way that absolutely meets the requirement of Lord Justice Leveson's report.
"That means million-pound fines, proper investigation of complaints, prominent apologies, a tough independent regulatory system. And they know, because I told them, the clock is ticking for this to be sorted out."
"That means million-pound fines, proper investigation of complaints, prominent apologies, a tough independent regulatory system. And they know, because I told them, the clock is ticking for this to be sorted out."
Labour leader Ed Miliband said he had spoken to victims of press intrusion, who want to see "more than good intentions" from newspaper editors.
Labour leader Ed Miliband said he had spoken to victims of press intrusion, who wanted to see "more than good intentions" from newspaper editors.
Speaking after meeting them, he added: "What those victims are saying is that we've heard those good intentions before but those good intentions have essentially drained away. We have ended up back to where we've started.
He said: "What those victims are saying is that... those good intentions have essentially drained away. We have ended up back to where we've started.
"They want to see real legislation which can ensure that we have independent self regulation guaranteed in law, so that other people don't find they are victims of abuse by the press."
"They want to see real legislation which can ensure that we have independent self-regulation guaranteed in law, so that other people don't find they are victims of abuse by the press."
Press Complaints Commission chairman Lord Hunt, who also attended the Number 10 talks, has been leading the industry's attempts to draw up a replacement for his own body.
An online petition in favour of statutory underpinning, launched by campaign group Hacked Off and supported by many victims of press intrusion, has collected more than 138,000 signatures.
He said he had the support of 120 publishers, representing 2,000 editors, for his plan for a new independent regulator.
Cross-party talks are ongoing and the government is drawing up draft legislation to enact the Leveson recommendations . Labour says it is drafting its own bill in case no agreement can be reached.
He said newspapers could sign legally enforceable membership contracts, which would mean it was not necessary to have their agreement backed up with new legislation.
But an online petition in favour of statutory underpinning, launched by campaign group Hacked Off and supported by many victims of press intrusion, has collected more than 135,000 signatures.
'Euphoria'
Cross-party talks are ongoing and the government is drawing up draft legislation on a new press watchdog.
He says some ministers hope a judge may lead the process of forming the new watchdog - "that, goes the argument, should silence those demanding legal underpinning".
He says some ministers hope a judge may lead the process of forming the new watchdog - "that, goes the argument, should silence those demanding legal underpinning".
Chris Blackhurst, editor of the Independent newspaper, told BBC Radio 4's Today programme he expected a very different tone from Mr Cameron than last week, when his response to the idea of legislation was greeted with "euphoria" in many newsrooms.
In Scotland, MSPs are also debating press regulation and whether to take a UK-wide approach or create a separate system.
He said: "I expect the tenor of today's meeting will be very different. He will be saying 'you have to get your act together and if you don't you will get statute'."
Mr Blackhurst said some papers, such as the Independent, Guardian and Financial Times, thought the new press watchdog needed to be more independent of the press.
But others, such as the Telegraph Group and the Daily Mail and Mail on Sunday, have in the past been "four-square" behind Lord Hunt's plan, he said.
Mr Blackhurst added: "There has got to be some movement. How long it takes I don't know. If it takes too long we will get statute."
Steve Hewlett, commentator and editor of the BBC's Media Show, told Today there was "a certain amount of theatre" about Tuesday's meeting, with the prime minister anxious to show he was getting tough with newspaper editors.
But he said most editors agreed that the status quo was not viable and had agreed a lot of what Lord Justice Leveson proposed - fines of up to £1m, full-page apologies and compliance. The sticking point, however, he said, remained the independence of the watchdog.
Mr Hewlett said there was still the threat of legislation if the editors did not agree to everything Mr Cameron demanded, but he added: "The cynical view is to say the press might hold on and hold on and nothing changes. But I think that is extremely unlikely."
David Banks, former editor of the Daily Mirror, told BBC Radio Five Live most journalists had been "horrified and chagrined" by the revelations which came out during the Leveson inquiry.
'Dark moment'
In a debate in the Commons on Monday, Mrs Miller told MPs the Leveson report marked a "dark moment in the history of the British press" and there was a need for cross-party unity.
She said: "Change can either come with the support of the press or - if we are given no option - without it. Be in no doubt that if the industry doesn't respond, the government will.
"We will not accept a puppet show with the same people pulling the same strings."
She said the government had "grave concerns" about the judge's call for press regulation to be underpinned by statute, but had not ruled it out.
Mr Miliband and Lib Dem leader Nick Clegg have voiced their support for the Leveson report, and Labour says it is drafting its own bill in case no agreement can be reached.
