One of the ads used an image of the House of Commons and asked readers, "Do you want them to decide what you read in The Sun?"
The campaign is being co-ordinated by the Free Speech Network – a body backed by organisations including the Advertising Association and The Newspaper Society.
Speculation is rife that Leveson will propose a regulatory system to replace the Press Complaints Commission (PCC), following calls from victims of phone hacking, including the parents of murdered teenager Milly Dowler, for an independent regulator backed by the law.
But newspaper publishers fear that such moves would stymie press freedom. Yesterday, The Sunday Telegraph, carried an ad asking, "If the press was shackled, would any of this ever have happened?"
Beneath the copy appears the front pages of six papers, including The Daily Telegraph, The Guardian, The Sun, the Daily Mail and the Liverpool Echo. Each cover bears a high-profile exclusive news story, including The Telegraph's breaking of the MPs' expenses story and The Guardian’s coverage of the phone-hacking scandal.
Underneath, more copy read: "Yes… even phone-hacking was a scandal revealed by a newspaper. Not by politicians, not by the police, and certainly not by a bunch of quangocrats.
"Say no to state regulation of the press."
The Free Speech Network ad campaign was reinforced by comments made by media and political figures appearing on TV yesterday, including Independent owner Evgeny Lebedev and foreign secretary William Hague.
Leveson’s report is expected to be published on Thursday, when the prime minister David Cameron is expected to respond within hours of it going live.
Yesterday, Downing Street repudiated the notion that it had already decided to reject the statutory implementation of a more rigorous form of self-regulation should Leveson propose it, although Cameron has said in the past that he would implement Leveson's recommendations provided they were not "bonkers".
However, the signs from Cameron's Cabinet are that the Government favours a free press. Speaking on BBC One's 'Andrew Marr Show' yesterday, Hague said the Government should "err on the side of freedom", adding that he wanted to read the report before he pronounced on it.
Hague was joined by London Evening Standard and Independent owner Lebedev. The Russian said that growing up in his homeland, where the press is over-regulated and state-controlled, meant that he was vehemently opposed to "any form of government regulation".
He added: "That said... I have got great sympathy with the victims of phone hacking, the families of the Dowlers and the McCanns.
"If we are to stay with some sort of self-regulation, it has to be extremely different from what it was before."
Meanwhile, writing in The Guardian today (26 November), the Labour leader Ed Miliband said that Cameron should show confidence in the Leveson report.
Miliband said: "Parliament set up this inquiry with the power to examine all the evidence and reach conclusions. We need to show confidence in this process, not try and invent a new one."