Lord Justice Leveson has rebuffed the government by making clear it was not his inquiry's role to rule if the culture secretary, Jeremy Hunt, has breached the ministerial code by his handling of the News Corp bid for BSkyB.

The firm refusal from the Leveson inquiry is embarrassing to David Cameron, who claimed on Wednesday that the inquiry was the best forum to determine whether Hunt, as well as his special adviser Adam Smith, had handled the bid in a partisan manner. Instead, Hunt may now have to face a separate, and potentially more painful, investigation by an independent watchdog set up to police the behaviour of ministers.

Leveson's spokesman also denied claims by the deputy prime minister, Nick Clegg, that he was going to bring forward the date of Hunt's appearance at the inquiry so his case could be fast-tracked. Clegg said: "I think we've already got an agreement Jeremy Hunt will go to the Leveson pretty quick." Leveson's spokesman said that Hunt's request to bring his evidence session forward had been turned down "in the interests of fairness to all".

Leveson's stance underlines that Cameron has not ordered an inquiry by the independent watchdog into the breaches of the code, but may now come under irresistible pressure to do so.

Ed Miliband had earlier accused Cameron of a cover-up. The Labour leader said the proper course would be to refer Hunt's case to Sir Alex Allan, the prime minister's independent adviser on the ministerial code. Sir Alex, who is paid £30,000 a year, has so far been excluded from the process on Cameron's instructions. His office was also bypassed by the prime minister over the conduct of Liam Fox, the former defence secretary.

Both the Lib Dem deputy leader, Simon Hughes, and Lorely Burt, who chairs the Lib Dem parliamentary party, called for Hunt to be referred immediately to Allan.

But Clegg, their party leader, said: "Unless anyone has got a better idea I think having a judge where a cabinet minister needs to give evidence under oath is about the best context to really get down to find out what happened or what didn't happen."

He added that after Hunt had given his evidence it might be possible to look at breaches of the code. Clegg, who denied that the coalition was sleazy, claimed there was a danger of crossed wires if Allan also looked at breaches of the code, such as ministerial responsibility for the conduct of a special adviser.

Leveson himself said on Wednesday after a telephone discussion with Sir Jeremy Heywood, the cabinet secretary: "Although I have seen requests for other inquiries and other investigations, it seems to me that the better course is to allow this inquiry to proceed."

But it is understood the Leveson inquiry is now concerned that it is being asked to examine issues of ministerial conduct in relation to the code well beyond its original terms of reference.

The inquiry believes that Lord Justice Leveson is not the arbiter of the ministerial code, because that falls to Allen.

Leveson sources were clearly anxious not to get locked into a row with Downing Street, but felt it necessary to assert their role.

Hunt said he would be handing to the Leveson inquiry all the texts and emails between himself and his special adviser over the bid, adding he was confident that they would show he had behaved with total integrity.

Cameron's plans to rely on Leveson's cross-examination of Hunt came under question from Sir Christopher Kelly, the chairman of the committee on standards in publiclLife. He said Allan was the obvious person to conduct an inquiry. If the issue were to be left to Leveson, Downing Street must make clear that the judge had the power to look into ministerial conduct – something that Cameron's official spokesman has repeatedly confirmed is a matter for the prime minister.

There was "no doubt" that the allegations needed to be properly investigated, said Kelly. "It is important for public confidence in the integrity of government and also in fairness to the individuals concerned that this is done – and done reasonably quickly. One obvious way to do it is by asking the independent adviser on ministerial interests to look at them. If it is to be done instead by Lord Justice Leveson as part of his inquiry then it needs to be clear that all the standards issues, including those relating to the ministerial code, are regarded as being within his remit and will indeed be looked at. It would be helpful to have that put beyond doubt."

A poll published on Friday night revealed that the public believes by a margin of five to one that Hunt should resign.

The ComRes poll for the Independent found that 63% of those polled believe Hunt should resign, 12% disagree and 24% don't know. The poll also gives Labour a five-point lead on 39%, the Tories 34%; the Lib Dems 10% and others 17%.

By a 2-1 margin, people think the government is incompetent. Only 27 per cent agree that it is proving competent. By a margin of 67 to 21 per cent, the public believe that David Cameron and George Osborne are out of touch with ordinary people.

Miliband, criticised by some in Labour for focusing too much on News International, kept up the pressure. He said: "Every day, David Cameron looks more like a prime minister organising a cover-up rather than standing up for the public.

"First he refuses to sack Jeremy Hunt despite the weight of evidence against him. Now despite all-party calls to do so, he refuses even to ask the independent adviser on ministerial interests to examine whether Mr Hunt broke the ministerial code.

"As Downing Street admits, it is not Lord Justice Leveson's job to adjudicate on whether Jeremy Hunt has broken the code.

"Just as last July, the prime minister dragged his feet on a judicial inquiry, defended Rebekah Brooks and clung to the BSkyB bid, so we see the same pattern again."

Harriet Harman, the shadow culture secretary, also wrote to the former cabinet secretary Lord O'Donnell to ask whether he would have cleared Hunt to act as the cabinet minister ruling on the BSkyB bid, had he known what has now been revealed about the conduct of Hunt's office.

Lord O'Donnell is expected to state his views when he appears in front of the inquiry himself.