Mr Cameron is opposed to enshrining the royal charter in law, saying it will harm press freedom.
Government sources told the BBC they prefer for the royal charter to be agreed without recourse to legislation.
The prime minister has indicated that he would want to take the royal charter to the Privy Council in May.
But much remains to be studied before the royal charter can be accepted as the foundation stone of new regulation.
The royal charter created in the wake of Leveson will create a so-called recognition body to oversee a new press regulator.
He added the Royal Charter would be in place to ensure the system was independent and not to exercise control itself.
Among the items in the archive is the Royal Charter of Carlisle from 1316, which bears the seal of Edward II.
The royal charter says it covers websites that provide news-related material, but there is some confusion as to what that really means.
Wootton Bassett Town Council has confirmed that any surplus money raised through donations to the royal charter celebrations will be donated to various forces charities.
Ms Hyslop told the committee her government would also like to see the royal charter amended to reflect concerns raised by Glasgow couple Margaret and Jim Watson.
The market, which was granted the Royal Charter in 1700, is visited by up to seven million people a year who use its 150 shops and food stalls.
The royal charter she maintained "reflects a principled way forward, proposed by the Conservatives from this side of the coalition and we're clear that it is a workable solution".
Deputy Labour leader Harriet Harman raised a number of issues with the proposals, but said the royal charter framework would form the basis of cross-party talks scheduled for Thursday.
"Rather as they did for the Royal Charter agreement after Leveson, Messrs Cameron, Clegg and Miliband should now sit down together and talk about this issue, " the Labour MP said.
This bill just happens to be in the House of Commons right now so that bit will be considered first by MPs, not peers who will look at the royal charter first.
After the debate, the convener of the education and culture committee, Stewart Maxwell, said the committee would be considering the implications for Scotland of the royal charter on the self-regulation of the press.
She raised concerns about the possibility of government ministers being able to tamper with the royal charter through the Privy Council at any time and urged clauses in statute providing this could not happen.
The Scottish government did not endorse the expert panel's findings and are now trying to get the royal charter proposed by the leaders of the main UK parties amended to make it comply with Scots law.
It's a little bit after the fact, given that the legislative changes for the Royal Charter system they agreed between them have already been voted through, but events have moved so fast that the committee is playing catch-up.
While the industry said it needed more time "to study" the proposals before it made a full response, its statement pointed out that early drafts of the royal charter contained "several deeply contentious issues which have not yet been resolved with the industry".
The debate will be a chance to consider the many responses to Leveson: the Letwin Royal Charter idea, the Lestor bill, the Labour proposals - and Labour peer Lord Lipsey, who'll be opening proceedings, has arranged for briefings from the Media Standards Trust and others.
The other main differences in the rival royal charter, which Mr Clegg launched with Mr Miliband, involve not giving the press a veto over the members of the regulator and preventing it from being watered down or strengthened by future governments.
The industry's proposal is closely based on the draft royal charter published on 12 February following negotiations with national and local newspapers and magazines.
BBC: Leveson report: Newspapers reject press regulation plans
But Prime Minister David Cameron said he did not believe a bill was necessary to set up the new regime and, instead, the Conservatives say a royal charter is the right way to provide legal backing for any new press regulator.
Imagine this brand freed from the restraints of Royal Charter, able to fundraise and fling itself into the market, to act as ruthlessly commercially as anyone else.
Mrs Miller has said she is reluctant to legislate but ministers have floated the idea of a Royal Charter to monitor the operation of the new press watchdog.
Neither government nor Parliament runs the BBC even though MPs do vote every few years to preserve the BBC by renewing its Royal Charter and set the level of the licence fee whilst the government appoints the Chairman of the BBC Trust which does regulate it.
Mr Llewellin said he preferred reform of the voluntary Press Complaints Commission code and described the idea of a royal charter as "perverse" in the Scottish context.
Industry sources told BBC political correspondent Ross Hawkins the industry-backed press regulator would not apply for formal recognition under the government's royal charter plans and it will not apply to be the official regulator.
BBC: Leveson report: Newspapers reject press regulation plans
He added that the idea of a royal charter was "not ideal" and "pretty meaningless" the way it was currently drafted.
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