Newspaper editors have been warned by David Cameron that they must act "urgently" to set up an independent press regulator.
The Prime Minister took part in a summit of senior industry figures at Number 10 to hear proposals for a new regime not backed by law.
Culture Secretary Maria Miller has made clear that the option of legislation called for by the Leveson Report remains an option if progress cannot be made.
Speaking after taking part in the high-level meeting, Mr Cameron said he had made clear that "the clock is ticking" for the industry to agree action.
The Prime Minister has expressed "serious concerns and misgivings" about resorting to any form of statutory underpinning for press regulation. But with Labour and the Liberal Democrats united in favour, his own backbenches split and phone-hacking victims leading a campaign for full implementation, he is under huge pressure.
Lord Hunt - who chairs the existing Press Complaints Commission - went into the talks claiming the support of 120 publishers, representing 2,000 editors, for a new independent regulator. He believes legally-enforceable membership contracts could be used to give the new system force and avoid the need for statutory backing.
Officials at Mrs Miller's Culture Department are drawing up a draft Bill to enact Leveson's recommendations in full, but she has indicated that she expects it to provide confirmation of concerns about the complexity and potential negative impact on press freedom.
Mrs Miller said the industry had "responded positively" to the challenge and would set out proposals within the next two days. Speaking after the talks, she said: "There was unanimous agreement that what we need now is for the press to go forward with developing an independent self-regulatory body. The challenge has been thrown down to them. They have responded positively and it is now for them to go away and develop those plans."
Daily Telegraph editor Tony Gallagher said the summit was attended by 19 editors and industry representatives, nine Whitehall officials and four ministers, including the Prime Minister and the Culture Secretary. "It felt like the summoning of the Five Families in The Godfather," he joked on Twitter.
Labour leader Ed Miliband, who met Hacked Off representatives including phone-hacking victims in his office as the summit went on in Downing Street, said they wanted "more than good intentions" from the press.