The prime minister, David Cameron, gives his reaction to Lord Justice Leveson's report into press ethics ITN

David Cameron is facing accusations of being too quick to dismiss the Leveson inquiry's proposals after the culture secretary confirmed the government would produce a draft bill of the judge's recommendations in an effort to demonstrate that statutory regulation of the press is unworkable.

Maria Miller said on Friday she would meet the deputy prime minister, Nick Clegg, and Ed Miliband, the Labour leader, next week as the government attempts to head off criticism from victims of press intrusion.

The father of Madeleine McCann expressed his concern over Cameron's swift dismissal of a key Leveson proposal, saying that statutory underpinning was the "minimal acceptable compromise" for press reform.

Speaking on BBC Radio 4's Today programme on Friday, Miller repeated her "grave concerns" about Leveson's proposed system of regulation enshrined in law.

"The prime minister was very clear: we have some grave concerns about the principle of putting in place statutory underpinning for this new body and we're not convinced it's absolutely necessary to achieve what he's set out," she said.

"What we're concerned about is creating amendable legislation that could in the future give parliament the opportunity of stopping reporting on certain areas."

Miller also raised concerns about Leveson's suggestion that Ofcom act as a "backstop regulator" of the press, because the culture secretary is responsible for appointing its chief executive. She said it was "surprising" that Miliband had backed all of the recommendations made in the report, but maintained that the government "fully accepts the principles" laid out in the 2,000-page document.

Miller refused to be drawn on whether Cameron believed the report to be "bonkers" – a reference to the prime minister's on-air pledge on BBC1's Andrew Marr programme in October that he would implement Leveson's recommendations unless they were "bonkers".

Earlier on Friday the deputy Labour leader, Harriet Harman, piled pressure on Cameron when she backed concerns expressed by victims of press intrusion, including the parents of Milly Dowler and the 7/7 survivor Prof John Tulloch.

She said: "What we're talking about is ensuring that that code is independently enforced. That is what Leveson has proposed, and for the sake of the victims that does not seem too much to ask."

Gerry McCann, the father of Madeleine, who went missing in 2007, said he was dismayed by Cameron's reaction, but also said he believed Leveson's report did not go far enough.

"Although we broadly welcome Lord Leveson's report and it has many merits, for me I don't think it has gone far enough. I would have liked to see a properly independent regulation of the press whereas I think he's given them another chance at self-regulation," said McCann.

Asked what he made of the prime minister's swift dismissal of the statutory underpinning proposal, McCann told the Today programme: "Full implementation of Lord Leveson's report is the minimal acceptable compromise for me and many other victims that have suffered at the hands of the press.

"I think the prime minister and our other elected politicians have an opportunity to do the right thing. I think if they do the right thing for the public then it will help restore a bit of confidence."

Alan Rusbridger, the editor-in-chief of Guardian News and Media, said he had spoken to editors on Thursday night and there were "minor differences" about whether statute was necessary.

"Our view is if you need a bit of statute to underpin that [incentives to join], to make it effective in the courts, that's something worth having. This big bogeyman of statute I don't think is there," he said.

The Sun, the Daily Mail and the Daily Mirror – the papers that had campaigned vociferously against any legislation – all heaped praise on Cameron in leader columns on Friday.

The veteran journalist and editor Max Hastings also voiced his opposition to Leveson's statutory recommendation. He told Today: "As soon as parliament legislates [that] the British press should be regulated then however you dress it up you have a degree of state regulation."

However, the former Sunday Times editor Harold Evans said self-regulation had been a "total failure" and he could not see the problem with "some kind of authority" behind it.

Miller will meet phone-hacking victims on Friday and will be pressed to explain why the prime minister earlier said he would implement the Leveson proposals so long as they were not "bonkers", but is now saying he is opposed in principle to any further involvement of the law in the underpinning of a new press regulatory body.

She will meet newspaper editors next week to impress on them that she must see signs of progress on their proposals for a new tougher independent regulatory body. But the industry is divided, with the Guardian questioning the wisdom of Lord Hunt, the chair of the Press Complaints Commission, remaining in place after his proposals developed over the past year were roundly rejected not only by Leveson but also by the entire political class on Thursday.

Labour accused Miller of prejudging the impracticality of a draft bill, accusing her of preparing to present laws in the next two weeks that would be so draconian that only the Stasi would welcome them. Labour insisted the legislation to underpin the new regulator could be done cleanly, simply and effectively.

Miliband has said he will wait until the end of January at the latest to table a motion calling for the Commons to accept statutory underpinning. On current maths, he should be able to win this vote but much will depend on developments between now and then, and whether a cross-party consensus can hold.