Newspaper editors will hold crunch talks today after David Cameron warned that "the clock is ticking" to avoid regulation backed by law.
The Prime Minister has told the press it must act fast to convince politicians and the public that Lord Justice Leveson's call for statutory underpinning is not necessary.
The push for an agreement comes as Labour, the Liberal Democrats and victims of media abuses continue to demand full implementation of the judge's recommendations.
Mr Cameron's own party is also seriously split on the need for legislation.
More top news
Son of barrister hired by Lord Janner told ITV News his father once believed peer would have to face charges over child abuse allegations.
Low pressure brings cooler, showery conditions over the next week
100-year-old tourist mugged by stranger after he fell getting off a bus as he returned to Gloucester where he used to live.