Culture Secretary Maria Miller has insisted the "principles" of Lord Justice Leveson's press regulation blueprint can be met without giving it statutory backing.
She dismissed the question of legislation - which has put David Cameron on a collision course with his Coalition partners, Labour and the victims of press intrusion - as "a matter of detail".
But the father of missing Madeleine McCann said legal backing for any new system was the "minimum acceptable compromise for me and for many other victims" and urged the Prime Minister to "do the right thing".
Lord Justice Leveson yesterday condemned the "culture of reckless and outrageous journalism" that dominated sections of the press for decades, as he unveiled the findings of his 16-month inquiry.
The Appeal Court judge called for a new watchdog with statutory underpinning to be given the power to require prominent apologies and impose fines of as much as £1 million.
Mr Cameron immediately voiced "serious concerns and misgivings" about legislative action, and said the press should be given "a limited period of time" to show it could get its house in order.
But Deputy Prime Minister Nick Clegg said he believed the Leveson model could be "proportionate and workable" and Labour leader Ed Miliband urged MPs to "have faith" in the proposals.
The three leaders met for initial talks last night but consensus appeared distant.
Draft legislation is being drawn up - but Ms Miller indicated that it was intended to illustrate how hard it would be to formulate such a change without threatening future press freedom.
Gerry McCann, whose family's treatment at the hands of the press following his daughter's disappearance were highlighted by Lord Justice Leveson, said he would have liked the report to go further.
"I would have liked to have seen a properly independent regulation of the press, whereas I think he has given the press another opportunity of self-regulation," he told the BBC Radio 4 Today programme.
It should be made compulsory and measures put in place make journalists more accountable, he said.
"But I do accept that full implementation of Lord (Justice) Leveson's report is the minimum acceptable compromise for me and I think for many other victims who have suffered at the hands of the press.
"The Prime Minister and our other elected politicians have an opportunity now to do the right thing. And if they do the right thing, for the public, then it will help restore a little confidence.
"I clearly respect his opinion but I personally disagree with the viewpoint and Lord (Justice) Leveson, as a senior law judge of our country, has made clear that what he is proposing is not a state-run press.
"It is a fine distinction but without the statutory underpinning this system will not work."
Ms Miller told the programme: "The horrific experiences of the McCanns and others cannot be allowed to happen again and that is why we are absolutely clear that we fully accept the principles of Leveson but, as you would expect, we would look at the details of implementation.
"We have some grave concerns about the principle of putting into place statutory underpinning for this new self-regulatory body but we are also not convinced that it is absolutely necessary to achieve the objectives that both Lord (Justice) Leveson set out and indeed the victims have set out."
She declined repeatedly to say whether she believed the statutory underpinning recommendation was "bonkers" - the test previously suggested by Mr Cameron for whether he would implement any proposed reforms.
"What we are absolutely doing is going to move forward with the core recommendation of the Leveson report, which is an independent self-regulatory body," she said.
"Actually the judge did say that it was a matter of detail. The prime focus of the judge's report yesterday was the importance of the press themselves to now come forward with a self-regulatory body framework that can work. That's what the McCanns and others want to see.
"What he wants to see is effective regulation. What we are saying is that we have grave concerns about making that effective regulation underpinned with statute.
"What we are concerned about is creating amendable legislation that could in the future give a framework which could give Parliament the opportunity of stopping reporting on certain areas.
"You have to consider that carefully before going forwards."
She said the draft legislation was being drawn up "to look at what that Bill might look like, to demonstrate our concerns".