Labour sources said the two parties were clear that any deal would have to be on the basis of its charter - which also beefs up the power of the watchdog to enforce prominent apologies. It also rules out any veto for newspapers over the membership of the regulator.
News of the revived talks came after both Chancellor George Osborne and Culture Secretary Maria Miller indicated the door was open to a compromise.
"Ultimately we're not about grandstanding on this. We're about getting a press law that works," Mr Osborne told BBC1's Andrew Marr Show.
Mrs Miller said she hoped both sides could "come together and have a real solution here" but warned statutory underpinning would have a "chilling effect". However while Number 10 sources said the statutory underpinning was "unnecessary and undesirable", Mr Cameron signalled it was not "a big issue of principle".
A senior Labour source said: "We are in lock-step with the Liberal Democrats on this. Ed Miliband spoke to Nick Clegg twice before Nick spoke to David Cameron and once after. We are clear we are not going to accept their royal charter. Any agreement must be on the basis of our royal charter. We are planning to go ahead with the votes in the Commons."