National newspapers have "pretty much agreed" to all the non-statutory elements of an independent new press regulation system, one editor said after industry talks.
The Independent's Chris Blackhurst said there had been "remarkable" unity across Fleet Street as they sought a sufficiently-tough package to see off calls for a legally-backed watchdog.
He expressed optimism over a possible deal after David Cameron told MPs he had been "encouraged" by a Downing Street summit on Tuesday where he warned editors the "clock is ticking".
The Prime Minister is reluctant to impose the statutory oversight recommended by Lord Justice Leveson's inquiry into press ethics, which was published last week.
He believes such a move could threaten free speech in future but he is opposed by Labour, the Liberal Democrats and many Tory MPs - as well as campaigners and victims of press abuse.
Mr 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", he said, notably over the practical operation of a proposed arbitration system and a non-statutory "verifier" proposed by ministers.
But there was general agreement that the new body should have the powers proposed by Leveson, such as the ability to impose fines of up to £1 million, launch its own investigations and dictate the size and prominence of apologies.
Mr Blackhurst said they supported in principle suggestions put forward at the 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.
The Hacked Off campaign group said any such solution would create "an entity with little authority invented by the Prime Minister and approved by the press".