Downing Street has insisted that it was approaching talks on the regulation of newspapers "in good faith", despite David Cameron's rejection of the Leveson Report's key recommendation of legal underpinning for a new press watchdog.

Cross-party talks on Lord Justice Leveson's report were resuming on Monday, with Culture Secretary Maria Miller meeting her Labour counterpart, Harriet Harman, and Liberal Democrat Lord Wallace, ahead of a major Commons debate on the highly divisive proposals for reform.

MPs will debate in detail the contents of Lord Justice Leveson's 2,000-page report later but will not have any opportunity to vote on it. The debate comes as an online petition calling for full implementation of the report, backed by campaign group Hacked Off and many victims of press intrusion, passed 125,000 signatures.

Following the report's publication last Thursday, Mr Cameron said he backed Lord Justice Leveson's plan for a new press regulator, truly independent from the industry and with the muscle to impose big fines and require prominent apologies.

But he clashed with Deputy Prime Minister Nick Clegg and Labour leader Ed Miliband over the Appeal Court judge's call for the body to be underpinned by legislation, telling MPs he had "serious concerns and misgivings" about the move.

In initial talks with Mr Miliband and Mr Clegg last week, the PM agreed to ask the Culture Department to draw up a draft Bill to introduce the judge's recommendations in full. But sources indicated that Mr Cameron expects the process to demonstrate the complexity of writing the plans into law, leading critics to speculate that the Bill may be drafted in such as way as to highlight potential difficulties. And no timetable has yet been announced for the completion of the task.

Labour on Sunday night upped the ante by revealing it had brought in a legal team to draw up its own Bill to counter claims that legislation would be "unworkable".

The Labour Bill will be used as the basis for a Commons vote before Christmas if cross-party talks fail to establish consensus. With both Labour and Lib Dems favouring statutory underpinning, any vote would carry the very real threat of defeat for Mr Cameron, but would not be binding on the Government.

Mr Cameron's official spokesman has dismissed suggestions that the Government's draft bill was merely a tool designed to make legislation appear impractical. "We are approaching these talks in good faith and we will draft the legislation accordingly," said the PM's spokesman. "The Prime Minister's view - which he set out in the House of Commons - is that it is likely that legislation will be quite complicated in practice. But we are progressing the work on that, and will continue the talks."

Draftsmen will be guided by the specific recommendations made by Lord Justice Leveson when writing the Bill, he said. On Tuesday, Ms Miller will host a summit with newspaper editors - to be attended by the Prime Minister - to discuss progress on establishing a new press regulatory body along the lines set out in the report.