The newspaper industry has announced its own proposals for a Royal Charter to underpin a new system of self-regulation.
David Cameron has insisted a cross-party agreement for a new system of press regulation protects the principle of the free press.
Labour says this is just the sort of legal under-pinning they were looking for but the Conservatives say that no such press law will exist.
Lord Justice Leveson says it would be quite wrong for him to comment on the politically contentious issue. So it is unlikely we are going to get his view on the way forward.
Lord Justice Leveson has arrived to give evidence in front of MPs.
Leveson admits there was always going to be strength of feeling about his report. He denies that his recommendations jeopardised the freedom of the press.
He seems a bit surprised at the Royal Charter idea, saying he didn't think of it and no-one suggested it at any stage.
Newspapers and magazines have begun to set up a new press watchdog in the wake of the phone hacking scandal - but have been instantly criticised by campaigners.
Plans for the Independent Press Standards Organisation, which will have the power to impose fines of up to £1m, will go out to consultation.
But Hacked Off, which represents some of the victims of phone hacking, claimed the move was a "cynical rebranding exercise" that showed the industry was "determined to hold on to the power to bully the public without facing any consequences".
John Prescott has resigned from the Privy Council in protest at a delay to new press regulation plans that "borders on a conspiracy".
The Labour former cabinet minister - a member of the Privy Council since 1994 - used his column in the Sunday Mirror to suggest the Government had deliberately "dragged its feet".
"I believe this approach borders on a conspiracy to delay Press regulation. Much worse, it will embroil the monarchy in a possible conflict with Parliament and political division between the parties."
The peer - who will no longer be entitled to be referred to as "the right honourable" as a result of his withdrawal - said he had only "reluctantly accepted" the cross-party "compromise" on the original Leveson recommendations.
"I sent off my resignation letter on Friday. The Privy Council must put Parliament and Parliament's Charter first.", Lord Prescott said.
The Government is to delay the presentation of its proposed Royal Charter to underpin a new system of press regulation to the Privy Council.
The charter had been due to be presented to the Queen by the Lord President of the Privy Council, Deputy Prime Minister Nick Clegg, for approval on May 15.
However, Downing Street said last night that it had been put back in order to give more time for consideration of an alternative charter put forward by the industry.
The move was welcomed by the Newspaper Society, representing the industry.
But the Hacked Off campaign said only the Government version would meet the requirements of the Leveson Report on press standards.
The alternative version - which has the support of most national, regional and local newspapers - is currently open for comment on the Privy Council website until May 23.
Prime Minister David Cameron has said he will listen to the newspaper industry's proposals for a Royal Charter that would create a new system of self-regulation:
The proposal for an alternative press regulator put forward by three major newspaper groups would, controversially, give newspapers the ability to veto appointments to the board, the Guardian reports.
This is likely to be a sticking point since the Labour party and campaign group Hacked Off believe that newspapers must not have so much influence over who runs the future regulator.
The Guardian reports the right of veto given to newspapers bosses was seen as one of the major weaknesses of the discredited Press Complaints Commission.
Proponents of the alternative regulator say appointments to the board would be "without any direction from industry or influence from Government".
The view from the Department for Culture, Media and Sport (DCMS) is that there is already a Royal Charter that has cross-party support and that the government remains committed to.
It does not think that newspapers should not be focussing on another one.
The emeritus editor of the Daily Mail Group, Peter Wright, has said he believes parts of the Royal Charter would be "unworkable".
In an interview on BBC Radio 4's The World At One, he explains why newspaper groups have set up their own proposal for a press regulator:
He also conceded that the Guardian, Independent and Financial Times newspapers have not yet "signed up on the specifics," but added that they thought it was "a good idea to get the ball rolling".