David Cameron has warned newspaper editors that "the clock is ticking" for them to avoid statutory regulation by agreeing on a tough new independent watchdog.
The Prime Minister demanded urgent action at a meeting with leading industry figures in Downing Street to discuss their response to the Leveson Report on ethics and standards.
Editors will meet again on Wednesday in a bid to agree proposals sufficiently independent and robust to persuade politicians and the public that legislation is not required to curb future abuses.
Culture Secretary Maria Miller said she expected them to come up with a process and timetable within two days - amid suggestions that detailed proposals could be ready within a fortnight.
But Labour leader Ed Miliband, who wants the Leveson proposals implemented in full, said "good intentions" would not satisfy victims of phone hacking and other intrusions.
Mr Cameron has expressed "serious concerns and misgivings" about resorting to any form of statutory underpinning for press regulation, as called for by Lord Justice Leveson. With Labour and the Liberal Democrats united in favour, his own backbenches split and phone-hacking victims fronting a public campaign for full implementation, he is under huge pressure.
Mrs Miller, who led the talks at Number 10, renewed a warning to editors at the talks that legislation would be the "only option" left to ministers if media chiefs failed to act quickly. She said the press had "responded positively" to the challenge but had to take responsibility for coming up with a credible alternative.
The editor of the London Evening Standard, Sarah Sands, said they were "talking about a couple of weeks to try to come up with a solution that really satisfies everybody".
Lloyd Embley, editor-in-chief of the Daily Mirror, Sunday Mirror and The People, said there was "a firm belief that papers can deliver Leveson principles far more quickly without legislation". That would be "better for public and free speech", he posted on Twitter.
But Hacked Off director Evan Harris, a former Liberal Democrat MP, said Mr Cameron and the editors were "deliberately missing the point". He warned: "Victims and their political supporters will not allow any sham Cameron-Murdoch pact to cloud the issue."