The prime minister's spokesman also confirmed that the terms of reference of the Leveson inquiry did not allow Lord Leveson to examine breaches of the code, as had been implied on Wednesday, but he still insisted Cameron saw no reason to refer the issue to Allen.
Labour has called for an investigation, and also urged Hunt to publish his phone records, emails and text conversations between him and his sacked special adviser, Adam Smith.
Jenkin said: "We have a new cabinet secretary, Sir Jeremy Heywood, and he should demonstrate his independence and advise the prime minister. If he thinks there has been a prima facie case of breach of ministerial code. It should go straight to the independent adviser who pledged to us in his evidence, Sir Alex Allan, that he would conduct swift preliminary enquiries to decide whether there was a case to be answered or not."
Labour leader, Ed Miliband, claimed it was not credible to say Smith acted as "a lone wolf" without the knowledge of Hunt in repeatedly briefing News Corp on how the culture secretary wanted to help the company in its bid for BSkyB.
The shadow culture secretary, Harriet Harman, accused Cameron of doing everything in his power to avoid an investigation: "Lord Leveson's inquiry is of huge importance, but it is not in its terms of reference to look into breaches of the ministerial code.
"That is the job of the independent adviser, and the prime minister will not allow him to look into this."
Simon Hughes, the Liberal Democrat deputy leader, added that he found it incomprehensible that an inquiry had not been mounted by the government into whether there had been a breach of the code.
It also emerged on Thursday that Heywood personally contacted Leveson late on Tuesday before Leveson advised that no other inquiries should be held into the Hunt affair.
Heywood's call to the judge came hours after Leveson's inquiry into media ethics published emails that suggested Hunt's special adviser had been systematically providing inside information to News Corp executives.
The Cabinet Office insisted Heywood had not been lobbying Leveson to make his statement discouraging other inquiries on Hunt's conduct. They said Heywood spoke to Leveson on Tuesday afternoon "to inform him of the fact that MPs were asking the cabinet secretary to investigate some of the allegations which were being made against Hunt and to seek his views which he has now made public".
Number 10 would not disclose whether Heywood rang Leveson following a discussion with the prime minister.
Leveson said at the outset of Wednesday's hearing: "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. When it is concluded, there will doubtless be opportunities for consideration to be given to any further investigation."
He added he was not seeking to circumscribe parliament.
It is not known if Allan is privately pressing the cabinet secretary to mount an investigation to see if there have been any breaches of the ministerial code.
Allan promised MPs on the PASC two months ago that he would not be the prime minister's poodle and suggested that where there was a media frenzy over the conduct of a minister he would be well-placed to conduct an investigation. He added he would resign if he believed he had been improperly sidelined.
The rules make it clear that the adviser on the code cannot initiate an investigation, and must await a reference by the prime minister.
Hunt also had a shaky morning when Jonathan Stephens, permanent secretary at the Department for Culture, Media and Sport (DCMS), told a select committee he would neither confirm nor deny his alleged role in allowing Smith to speak to James Murdoch's office. Later, the DCMS issued a statement saying Stephens was aware and "content" with Smith's role. It follows Hunt's claim in parliament on Wednesday that Stephens had "agreed" that Smith would be the point of contact with News Corp.
Appearing before the public accounts committee (PAC), Stephens was asked by the committee's chair, Margaret Hodge, whether he knew Smith was acting as a channel of communication between the department and Murdoch's office. "He [Hunt] has made it clear that he's providing full written evidence and is looking forward to providing oral evidence to the Leveson inquiry. There was a statement by the special adviser yesterday that made it clear he accepted that the nature and content of those contacts was not authorised by the secretary of state or by me and I think that that is the right forum for those matters," said Stephens.
He was then asked another nine times to clarify his role in approving Smith's position as a go-between, but declined to do so. At one point, Labour member Nick Smith accused him of "stonewalling". Hodge said she was astonished that the secretary of state had claimed that Stephens had approved Smith's role. "It just rather takes me aback from all my experience as a minister that you would have approved a special adviser to act as a channel of communication between one party and the department when the department and SoS [secretary of state] was playing this semi-judicial role. It just seems so inappropriate.
"I think clearly the secretary of state has to answer for his actions and I understand that under the ministerial code he is responsible for the conduct of his special adviser. But in this instance the secretary of state chose to tell parliament that you had specifically approved the role and that astonishes me."
