Jeremy Hunt, the culture secretary, has begged the Leveson inquiry to give him a chance to salvage his reputation after emails released by News Corp appeared to show that Hunt and his office passed confidential and market-sensitive information to the Murdoch empire to support its takeover of BSkyB.

Facing calls from the Labour leader Ed Miliband to resign, Hunt urged Lord Justice Leveson to change his hearings timetable and give him a chance to clear his name.

As the day-long questioning of James Murdoch ended, Hunt rushed to a meeting with David Cameron and the cabinet secretary, Sir Jeremy Heywood, to explain the emails and texts that appeared to show he ignored his commitment to MPs to act in a quasi-judicial and impartial capacity over the £8bn bid, one that only failed in the wake of the Milly Dowler phone-hacking furore.

In the most dramatic day of hearings at Leveson yet, the inquiry was shown emails written by James Murdoch's chief lobbyist, Frédéric Michel, written the day before Hunt was due to make a market-sensitive statement to parliament, which appeared to indicate that he was minded to approve the bid in negotiation with News Corp. The statement came just a couple of days after the former News of the World editor Andy Coulson was forced to resign as communications chief at No 10 amid mounting hacking allegations.

Michel told Murdoch at 3.21pm on Monday 24 January, with stock markets in London and New York open, that he had "managed to get some infos [sic] on the plans for tomorrow (although absolutely illegal!)" which set out the timetable of Hunt's announcement due next morning and quoted from the planned announcement. Hunt said that he wanted to negotiate with News Corp over potential undertakings in lieu (UIL) of a referral to the Competition Commission. Murdoch said on Tuesday that the reference to "illegal" material was a joke.

A day earlier, Michel explained in an email to Murdoch what he understood to be the culture secretary's true thinking behind his public remarks. "His view is that once he announces publicly he has a strong UIL, it's almost game over for the opposition … He very specifically said he was keen to get to the same outcome and wanted JRM [James Murdoch] to understand he needs to build some political cover on the process."

In another email, Michel reported a request from the Hunt team that News Corp "try and find as many legal errors as we can" in the Ofcom report on the public interest implications of the bid. When Hunt cancels a meeting with James Murdoch in November 2010 because he has received "very strong legal advice not to meet us today as the current process is treated as a judicial one", Michel reports to his boss that he can still talk to him on his mobile phone "which is completely fine".

James Murdoch confirmed on Tuesday that Hunt had called him following this exchange.

The emails were handed over by News Corporation on the order of the Leveson inquiry in a 163-page disclosure that had the effect of deflecting attention from James Murdoch's testimony, and even threatening to overshadow Rupert Murdoch's evidence on Wednesday.

James Murdoch told Leveson that Michel was simply "doing his job" in trying to get the bid approved, and that in any event he took all communications with politicians with a "grain of salt". Hunt was co-operative because "he didn't want to take any heat alone" and that "I have never met a politician who did".

The emails appear to show how News Corp expected Hunt to push for the BSkyB deal to be approved. After speaking to Hunt or a member of his team "before he went in to see Swan Lake" in February 2011, Michel told his boss: "I told him he had to stand for something ultimately … and show he had some backbone." Cameron's spokeswoman robustly defended Hunt, saying he had acted throughout on the basis of independent advice, in a process Hunt repeatedly described as "quasi-judicial". But Robert Jay, the QC for Leveson, suggested to Murdoch that the communications revealed amounted to the judge in a case "telling you behind the scenes ... that you're going to win".

In a statement on Tuesday, Hunt pleaded for time, saying: "Now is not a time for kneejerk reactions. We've heard one side of the story today but some of the evidence reported meetings and conversations that simply didn't happen. Rather than jump on a political bandwagon, we need to hear what Lord Justice Leveson himself thinks after he's heard all the evidence.

"Let me be clear: my number one priority was to give the public confidence in the integrity of process. I asked for advice from independent regulators – which I didn't have to do – and after careful consideration I followed that advice to the letter. I would like to resolve this issue as soon as possible which is why I have today written to Lord Justice Leveson asking if my appearance can be brought forward. I am very confident that when I present my evidence the public will see that I conducted this process with absolute objectivity and scrupulous fairness."

His statement implies he believes he has Cameron's support to cling to office for months including through the Olympics since Justice Leveson is unlikely to report in the short term.

Hunt was resting his defence partly on the admission by Michel that some of his internal emails referring to contacts with JH were in fact "no more than shorthand for what I was told by someone within Jeremy Hunt's office, almost invariably his special adviser Adam Smith".

But Michel states: "His advisers were there to assist and advise Jeremy Hunt and it was my understanding that when they told me something, it was always on behalf of the minister and after having conferred with him."

Hunt's aides were refusing to blame Smith, or accuse him of acting as a rogue operator, suggesting Hunt may have been aware of the content of these contacts.

But Hunt's aides insisted the culture secretary will give evidence under oath and reveal his phone records.

Miliband said: "Hunt should have been standing up for the interests of the British people. In fact it now turns out he was standing up for the interests of the Murdochs. He himself said that his duty was to be transparent impartial and fair in the BSkyB takeover. But now we know that he was providing advice guidance and privileged access to News Corporation. He was acting as a back channel for the Murdochs."