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
The Met Office has weather warnings in place for snow and ice until 10am in the north and west.
The plan, which would effectively strip banks of the benefit of recent corporation tax cuts, forms part of the party's election manifesto.
Pop star went into the icy waters of Chicago's Lake Michigan at a fundraiser for the Special Olympics and later said "her weave froze".