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".
More top news
A court has ruled that the original coffin in which Lee Harvey Oswald was buried belongs to his brother, not the undertaker's that sold it.
Manic Street Preachers lyricist went missing nearly 20 years ago but mystery still surrounds exactly what happened to him.
Deputy Prime Minister admitted he wanted to hit David Cameron on more than 20 occasions on late-night chat show 'The Last Leg'.