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".
The managing editor of The Sun, Richard Caseby, has launched a stinging attack on the government-sponsored Royal Charter, accusing it of being "draconian" and "bodged together".
An earlier deal for state regulation was bodged together by politicians and the pressure group Hacked Off at a secret late night meeting. The Prime Minister was asleep in his bed at the time.
This was not the independent, self regulation recommended by the Leveson Inquiry. It was rushed; it was draconian; it was it was a mess; and it is being condemned by commentators the world over.
It meant that the state would ultimately have the final say in what newspapers write, and it went further.
It basically blackmailed publishers into joining up because it threatened them with punitive damaged which could easily put them out of business.
Hacked Off member Dr Evan Harris says this proposal for press regulation is essentially media barons sticking two fingers up to the public, readers Parliament and Lord Justice Leveson.
The acting editor of the Times, John Witherow, has said his newspaper supports the alternative blueprint for press regulation because the government-sponsored Royal Charter "infringes too much on the public's right to know and on press freedom".
He explains his views in a video posted on the websites of The Times and News International.
The editors of newspapers in the News International stable have released a series of videos explaining their vision of a press regulator independent of the government.
The editor of the Sun is quoted on the News International website saying:
Sun readers expect journalists to behave responsibly, but don’t want them censored by a state-sponsored Ministry of Truth. This constructive proposal would create a tough but independent regulator supported by the vast majority of the industry – a workable solution which should command public confidence.
The editor of the Daily Telegraph Tony Gallagher described the government-proposed approach to press regulation in colourful language on Twitter:
Can anyone possibly be surprised we have rejected Lab-Lib-Hackedoff stitch up- not forgetting walk on role for sleepy Letwin.(PM was in bed)
The newspaper industry said the Royal Charter published by the Government in March has been condemned by a range of international media freedom organisations and enjoys "no support within the press" in the UK.
A statement co-ordinated by the Newspaper Society said: "A number of its recommendations are unworkable and it gives politicians an unacceptable degree of interference in the regulation of the press."
The industry's proposal is closely based on the draft Royal Charter published on February 12 following negotiations with national and local newspapers and magazines.
The statement described it as "a workable, practical way swiftly to deliver the Leveson recommendations, which the industry accepts, without any form of state-sponsored regulation that would endanger freedom of speech".