A petition calling for the UK's three main party leaders to bring in a new press watchdog backed by law has attracted more than 56,000 signatures since being launched by high profile victims of media intrusion on Friday.

Gerry McCann, father of missing Madeleine, and Christopher Jefferies, the landlord wrongly arrested for the murder of Joanna Yeates, urged the public to back their online campaign after David Cameron indicated he planned to spike recommendations to introduce legislation.

The Prime Minister's position has been roundly welcomed by the industry but sparked fury among victims and campaigners, with Harry Potter author JK Rowling the latest to wade into the row saying she felt "duped and angry" by the Tory leader's stance.

Culture Secretary Maria Miller will meet the newspaper industry's most powerful editors next week to push for urgent action in setting up a new press watchdog, hoping it will prove to the public new laws are not needed.

The Tory minister will warn Fleet Street it must not drag its feet when it comes to acting on Lord Justice Leveson's calls for it to devise an independent regulatory body, which the Conservatives believe can function properly without legislation.

Lord Hunt of Wirral, chairman of the Press Complaints Commission, who will also attend, told The Times he wanted a speedy industry resolution to help persuade the public and MPs laws were not needed to underpin the new regulator. "There's an awful lot we can agree on and I have suggested to the industry (that we) all read the report, digest it and seek out the common ground and unite with one voice," he said.

Ms Miller's warning will come the day before editors on the Code of Practice Committee, chaired by Daily Mail editor Paul Dacre, meet to decide how to tackle the Leveson recommendations.Mr Dacre said: "Lord Justice Leveson has set us a number of challenges: our task is to address them as urgently as possible."

But Mr Cameron is under intense pressure to drop his opposition to a law backing up the new watchdog the press have been tasked with devising. As well as a backlash from victims the report has ratcheted up Coalition tensions with the Liberal Democrats saying they are prepared to back the regulatory system set out in the Leveson report.

Lord Justice Leveson 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 earlier this week. 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. The Prime Minister will attend the meeting Ms Miller is holding with editors next week, Downing Street confirmed.