The Sun's political editor Tom Newton Dunn has tweeted:
The FT, Guardian and Indy have not signed up to rival Royal Charter - but want to keep talking.
The newspaper industry said its counter proposals on press regulation would protect "freedom of speech."
This Royal Charter proposal will deliver on Leveson and bind the UK's national and local newspapers and magazines to a tough and enduring system of regulation - tougher than anywhere else in the Western world - which will be of real benefit to the public, at the same time as protecting freedom of speech.
The newspaper industry firmly rejected the Government's Royal Charter proposal on press regulation and published its own plans.
A statement co-ordinated by the Newspaper Society said the new proposal has "widespread backing across the industry" and would deliver:
- Tough sanctions - with the new regulator able to impose fines of up to £1 million for systematic wrongdoing.
- Full and prominent correction of inaccuracies.
- Strong investigative powers enabling the new regulator to investigate wrongdoing and call editors to account.
- Genuine independence from the industry and from politicians, with all the bodies making up the new regulator having a majority of independent members appointed openly and transparently.
- Public involvement in the framing of the Code of Practice which binds national and local newspapers and magazines.
The newspaper industry rejected the Government's plans for the future of press regulation and published its own proposal for a Royal Charter to create a new system of self-regulation.
Newspapers are about to set out their response to proposals on press regulation.
Barristers David Sherborne and Carine Patry Hoskins reportedly went on holiday together to Santorini in August, days after the public hearings concluded and months before the inquiry ended.
But they are said to maintain that their relationship did not begin until after the Leveson Report was published in November.
Reports suggest Mr Sherborne, 44, and Ms Patry Hoskins, 40, did not tell Lord Justice Leveson that they had become close and he only learnt of their trip to Santorini in recent days.
Carine Patry Hoskins became known as the "woman on the left" in reference to her regular seated position during the Leveson Inquiry's televised hearings.
She served as the junior counsel in the Leveson team led by Robert Jay QC.
Barrister David Sherborne represented phone-hacking victims at the Inquiry, most notably Hugh Grant, and questioned Daily Mail editor Paul Dacre on the newspaper's coverage of the actor.
Lord Justice Leveson's letter responded to Conservative MP Rob Wilson, who had demanded the judge examine the relationship between Carine Patry Hoskins and David Sherborne.
Mr Wilson wrote to the Bar Standards Board requesting an investigation into the impact of the barristers' private relations on the role both played during the inquiry.
Lord Justice Leveson said it was "a matter for the Board to decide, what, if any, action to take", adding: "In the circumstances, I do not comment further."
But he unequivocally defended the role of Ms Patry Hoskins, who he said had "absolutely no input into any conclusions" reached by the inquiry.
Lord Justice Leveson has rejected claims that a previously undisclosed relationship between two of the barristers in his inquiry into press freedom threatened the legitimacy of its subsequent report.
The judge has said today in a letter that there was "simply no room" for a "breach of confidence or other conspiracy" as a result of personal relations between Carine Patry Hoskins, the inquiry's junior counsel, and barrister David Sherborne.
The Government has tabled an amendment in the House of Lords to exclude "small-scale" blogs from a new scheme of press regulation.
The short amendment, put forward by Home Office minister Lord Taylor of Holbeach, is expected to be debated tonight when peers discuss additions to the Crime and Courts Bill that would allow for exemplary damages and the awarding of costs against publishers who choose not to join a new regulator.
Lord Taylor's amendment would add "a person who publishes a small-scale blog" to the list of people and organisations who will be exempted from the new regulatory regime.
The list already includes bodies such as the BBC, book publishers and people who produce business newsletters.