Newspaper and magazine publishers have written to Culture Secretary Maria Miller pledging to support the major principles outlined by Lord Justice Leveson in his report on press reform, apart from the one recommending statutory regulation.
They said they were committed to establishing a new system of independent self-regulation and that work was under way on a draft contract.
The principles to underpin the new regulator include independence of appointments and funding, a standards code, an arbitration service, a speedy complaint-handling mechanism and the power to demand up-front, prominent apologies and impose million-pound fines.
Lord Hunt, chairman of the Press Complaints Commission (PCC), is to act as a point of contact between publishers and the government.
Although Leveson recommended a statutory body to oversee a beefed-up watchdog, Cameron is reluctant to take that step, warning it could pose a future threat to free speech.
However he gave the industry a chance to come up with its own beefed-up proposal to convince politicians and the public that statutory regulation not necessary.
The letter to Mrs Miller was signed by the Newspaper Society, the Newspaper Publishers' Association, the Scottish Newspaper Society, the Professional Publishers' Association and the Press Standards Board of Finance.
It said: "A number of groups within the industry, including national editors, have held further meetings. However, as you will be aware, publishers are responsible for the establishment and funding of the new system, and will be the signatories to the contracts that will underpin it.
"We can confirm that we are committed to establishing a new system of independent self regulation in accordance with the five Leveson Principles outlined by the Prime Minister and have agreed that Lord Hunt should be the point of contact between publishers and the government.
"We accept the clear majority of Lord Justice Leveson's main recommendations, although we still have legal work to undertake on a very small number of areas including principally appointments, confidentiality of sources, allegedly discriminatory reporting and funding structures. This will be completed by next week.
"We are also taking forward urgently the recommendations from Lord Justice Leveson on the provision of an arbitral arm to the new regulator."
A working group including lawyers, editors and senior industry executives is co-ordinating the development of a draft contract and regulations and will report to Lord Hunt "as soon as is practical", the letter added.
Lord Hunt said this week he hoped to see a new form of press self-regulation set up and at work by early next year.
He said he would expect the new form of self-regulation to have an early "opportunity to prove its independence" and prove a body underpinned by statute was not needed.
He said: "This body, I hope, will be created, will be tested pretty swiftly in its existence, and it will be for others to judge".
Cross-party talks on press regulation broke up yesterday with members no closer to an agreement - the key sticking point being the proposal for the watchdog to be backed up in law, but there were also differences over data protection reforms.
Lord Hunt said today that he had appointed three independent advisers to help him "adjudicate" on the industry's proposals for a new regulator.
They are Lord Phillips of Worth Matravers, the recently retired president of the Supreme Court; Guardian columnist and former editor of The Times Sir Simon Jenkins; and Lord Smith, the former Labour culture secretary and now chairman of the Advertising Standards Authority.
Lord Hunt said he wanted the industry's implementation committee to produce its proposal for the new regulator to be considered at a meeting next Thursday (20th) of about 60 newspaper and magazine representatives.
A further meeting will be held on January 10 to "maintain the momentum"
"The industry wants to endorse Leveson. There is a window of opportunity in which to do that.
"I believe newspapers and magazines in print and online now accept that there is no third way.
"I want to see a Leveson-compliant model."
He and the other two remaining directors of the PCC - Michael Grade and Peter Wright - will also be working on the "orderly closure" of the PCC and the transfer of some functions to the new body "as soon as possible".