David Cameron is prepared to consider ordering an investigation into Jeremy Hunt's dealings with News Corporation but not until the Culture Secretary has appeared at the Leveson Inquiry.
The Prime Minister has been resisting demands to call in his independent adviser on ministerial conduct, Sir Alex Allan, insisting it is a matter for Lord Justice Leveson's inquiry into media standards.
However, he is understood to be open to the possibility of a separate inquiry into whether Mr Hunt broke the ministerial code after the Culture Secretary has defended himself in front of Lord Justice Leveson.
The deputy chairman of the Conservative Party, Michael Fallon, said today that the evidence should be "tested" at the Leveson Inquiry.
"If later on, obviously, it looks as if there's something that needs to be investigated under the ministerial code that can be done," he said.
It emerged last night that Lord Justice Leveson would not rule on whether Mr Hunt broke the ministerial code.
The Culture Secretary has been accused of acting as a "cheerleader" for News Corporation's BSkyB takeover bid after the Leveson Inquiry published a raft of email exchanges between the company's chief lobbyist and Mr Hunt's office.
Mr Hunt's special adviser, Adam Smith, resigned over the matter this week.
The Culture Secretary is now facing a lengthy wait to defend himself at the Leveson Inquiry after his request for an early appearance was rebuffed.
Lord Justice Leveson has refused to bring forward his appearance so that he can answer allegations about his conduct.
A date has still not been set, but politicians will not be called until mid-May, meaning the Culture Secretary will have to wait at least a fortnight and possibly much longer.
The delay is a setback for the under-fire minister, who has expressed confidence he would be able to show he acted with "scrupulous fairness" when he sets out his full version of events to the inquiry.
He told MPs this week - amid claims that he was a "cheerleader" for News Corporation's attempt to take full control of BSkyB when he was supposed to be acting "quasi-judicially" - he had requested the earliest possible date to do so.
But a spokesman for the inquiry said: "Lord Justice Leveson is of the view that, in the interests of fairness to all, the inquiry should continue with the existing scheduling of his appearance."
But Labour leader Ed Miliband said the Prime Minister should have fired Mr Hunt and at the very least should refer the matter urgently to Sir Alex Allan, to protect the "integrity of his Government".
Speaking in Ealing Broadway, west London, where he met potential voters with mayoral candidate Ken Livingstone, he said: "Lord Justice Leveson has made now very clear that it's not a matter for him whether Jeremy Hunt broke the ministerial code.
"Therefore the Prime Minister can no longer hide behind Lord Justice Leveson.
"At the very least he's got to refer the matter to the independent adviser on ministerial interests.
"I think Jeremy Hunt should go, I think the Prime Minister should have fired him. But the very least he's got to do is refer it to Alex Allan.
"We've had Conservatives calling for that, we've had Liberal Democrats calling for that, we've got Labour calling for it - this is now an all-party issue.
"Frankly the longer the Prime Minister goes on resisting what seems to be the obvious thing to do, the more people will conclude he has something to hide, and he doesn't want the truth to be got at.
"I do say to him that as a matter of the integrity of his Government he now must refer this to Alex Allan at the very least.
"Urgently, now, he's got to do it, and he should have done it days ago.
"I thought he should have got rid of Jeremy Hunt. Given the revelations there were by Tuesday evening my view was that Jeremy Hunt should have gone.
"But given that he's chosen not to do that, the very least he's got to do is refer this matter to Alex Allan.
"After all, Alex Allan's role as independent adviser on ministerial interests is to look precisely at these issues and there are at least three ways in which Jeremy Hunt appears to have breached the ministerial code."
He said the longer the Prime Minister resisted, the more people would think he was involved in a "cover-up", and said: "there's no hiding place for him".
Mr Miliband added: "The integrity of his Government and the view people take about his Government depends now on the truth coming out about Jeremy Hunt, about what he did, and about what other members of Parliament, including the Prime Minister and the Chancellor, did about BSkyB."