Labour leader Ed Miliband has said "it beggars belief" that Culture Secretary Jeremy Hunt is still in his job.
Mr Miliband said the "accumulation of evidence" about Mr Hunt's links with News Corp when he was considering its BSkyB bid made his position untenable.
"He was standing up for Rupert Murdoch when he should have been standing up for the public interest," he said.
Senior Conservative Michael Fallon said it was "ludicrous" to judge Mr Hunt before his Leveson Inquiry evidence.
The Conservative deputy chairman said it was "very clear" that Mr Hunt believed he had handled the situation fairly.
"It is just ludicrous for Labour - or anybody else - to prejudge this on the basis of just one email before he's even given his evidence in front of the inquiry," Mr Fallon said.
Mr Hunt is resisting calls to quit, saying he has "behaved with integrity".
Mr Miliband said more evidence of Mr Hunt's links to Mr Murdoch were revealed at the Leveson Inquiry into press ethics on Friday.
The inquiry heard suggestions that Mr Hunt sought "private advice" from News Corp over its involvement in phone hacking.
"I have to say I think the government just keeps showing how out of touch it is - that they're clinging onto him," Mr Miliband said.
Shadow business secretary Chuka Umunna told the BBC that the government needed to address the allegations against Mr Hunt otherwise "they're not going to be able to bring the focus and clarity that we need to see".
He said: "The economic situation is pressing and urgent - that is where we need ministers' and the prime minister's focus to be not on this issue."
But Conservative Peer Lord Fowler told the BBC News Channel the allegations against Mr Hunt needed to be examined by the Leveson Inquiry before any decision on his future could be made.
"The fact is that allegations have been made, there is a case to answer, but we've got to listen to his defence - and we haven't heard his defence. I mean Labour is starting to sound like some sort of lynch mob," he said.
Deputy chairman of the Conservative Party Michael Fallon said it was "very clear" that Mr Hunt believed he had handled the situation fairly.
"It is just ludicrous for Labour - or anybody else - to prejudge this on the basis of just one email before he's even given his evidence in front of the inquiry."
Mr Miliband's comments followed a speech in London in which the Labour leader launched a fresh attack on the government over its past contact with News Corp.
Mr Miliband said the government had "bent over backwards" to help Mr Murdoch's media empire following the latest revelations from the Leveson Inquiry.
He also said Labour must "seize this moment" after last week's election, which saw his party gain hundreds of seats at the expense of the two coalition parties.
But Mr Miliband told the Progress annual conference Labour first needed to regain the trust of an electorate that had lost confidence in politicians.
He said Labour had also been "to close to the powerful interests" when it was last in power.
An email referred to on Friday at the Leveson Inquiry, sent to ex-News International chief Rebekah Brooks, said Mr Hunt wanted "guidance".
The message, from News Corp public affairs executive Fred Michel to Mrs Brooks, dated 27 June 2011, predicted correctly that Mr Hunt would play down the influence of the phone-hacking scandal on the company's bid for full control of BSkyB.
The message stated that Mr Hunt wanted to "prevent a public inquiry".
Mr Hunt has said the email was "inaccurate".
And a spokeswoman for the culture secretary said: "Jeremy Hunt will respond to this when he gives evidence to the Leveson Inquiry in due course.
"He is confident his evidence will vindicate the position that he has behaved with integrity on every issue.
"It has already been made clear that when Fred Michel has claimed in emails to be speaking to Jeremy Hunt that was not the case.
"On July 11 2011 Jeremy Hunt wrote to Ofcom for further advice about the impact of phone hacking on the BSkyB bid."
Mr Hunt's officials point out that Mr Michel has already admitted that the only contact he had was with the culture secretary's adviser, Adam Smith, who resigned last month.
Mr Hunt himself has said he "strictly followed due process" during the bid process and denied that News Corp had any "back channel" of influence with his office.
Downing Street has said there were "no plans" to investigate whether Mr Hunt had broken the ministerial code, with Mr Cameron saying the culture secretary had acted properly and that it was a matter for the Leveson Inquiry.