Ministers will seek commitments from newspaper editors today about progress towards agreeing a new independent press watchdog.
Culture Secretary Maria Miller set a two-day deadline for a timetable of action when she met senior Fleet Street figures in Downing Street on Monday.
National editors met yesterday to discuss joint efforts to produce proposals sufficiently strong to see off demands for a watchdog backed by law.
Lord Justice Leveson's inquiry into press ethics, prompted by the phone-hacking scandal, recommended a statutory body to oversee a beefed-up watchdog.
Prime Minister David Cameron is reluctant to take that step, warning it could pose a future threat to free speech, but has warned the industry it must act fast to convince politicians and the public that it is not necessary.
Labour, the Liberal Democrats and a large number of Conservative MPs are united with victims of press abuses and other campaigners in demanding full implementation.
Independent editor Chris Blackhurst said editors were "pretty much agreed" on all the non-statutory elements of the system proposed by the Leveson report.
That includes the power for a new regulator to levy fines of up to £1 million, to launch investigations and to dictate the size and prominence of apologies.
Blackhurst suggested there was also consensus on making the new body independent of the press - though with the caveat that journalists help draft a new code of practice.
But he said there remained "one or two grey areas" notably over the practical operation of a proposed arbitration system and a non-statutory "verifier" proposed by ministers.
Blackhurst told BBC Radio4's Media Show the meeting went through the Leveson proposals point by point "and pretty much agreed to them all".
There were still "one or two grey areas" notably over the practical operation of a proposed arbitration system and a non-statutory "verifier" proposed by ministers.
Blackhurst said they supported in principle suggestions put forward at yesterday's summit by Cabinet Office Minister Oliver Letwin for a non-statutory oversight body but that the details remained "incredibly vague".
"Him and his bright boys and girls across Whitehall have identified a body, an office, that might oversee us. We don't know who it is.
"It is incredibly vague. We are really in the dark," he said.
Proposals floated by Cabinet Office Minister Oliver Letwin at Monday's Downing Street summit for a non-statutory "verifier" remained "incredibly vague".
The Prime Minister told MPs yesterday that he had been "encouraged" by the talks that editors could come-up with "Leveson-compliant regulation".
"We should continue the cross-party talks and make sure we can deliver a regulatory system of which this House, this country, but, above all, the victims can be proud," he said.
But the Hacked Off campaign group hit out at what it said were plans for "an entity with little authority invented by the Prime Minister and approved by the press".
"Until the Prime Minister and Culture Secretary understand that the public stand with the victims in demanding that they stop interfering with what Lord Justice Leveson has set out to ensure the press cleans up its act and stays cleaned up, they will be seen as simply engaging in an attempted stitch-up with their cronies in the press," associate director Evan Harris said.