Mr Clarke told BBC News: "He should instruct his employee, Andy Coulson, to answer the questions that everybody wants to know following those revelations yesterday. Namely, were the News of the World bugging people? Did he authorise it? Did he pay for it? What did he know?
"So far we have had no answers from Andy Coulson. We need answers."
But Mr Cameron said Mr Clarke was indulging in "a political stunt".
"Labour are desperately trying to deflect attention away from the dreadful state of the public finances and the complete shambles of their government," he said.
He added: "The one thing Charles Clarke and I do agree about is that Gordon Brown is not the right man to be prime minister."
He said Mr Coulson had taken responsibility for what happened "on his watch" at the News of the World, when he resigned as editor two and a half years ago after Royal reporter Clive Goodman was jailed for phone hacking.
"I thought it was entirely reasonable for me, him having resigned, having done the right thing, to actually give him a second chance in a second job and all I can say is that as director of communications, he has behaved in an entirely proper way at all times, working for me and the Conservative Party."
The police have said they will not re-open their inquiry into the affair.
The Metropolitan Police said no new evidence had come to light to warrant the move despite allegations in the Guardian that up to 3,000 high profile figures - including John Prescott - were targeted by the newspaper.
But the Lib Dems are dissatisfied with this and have asked the Independent Police Complaints Commission to examine the conduct of the original inquiry.
The possibility that other journalists and investigators were involved must now be seriously considered
Chris Huhne, Lib Dem home affairs spokesman
In a letter to the Commission, the party's home affairs spokesman Chris Huhne said the Guardian allegations have "serious implications" for privacy laws and press freedom.
"Given the scale and scope of the allegations, the possibility that other journalists and investigators were involved must now be seriously considered," he said.
The Lib Dems have called for Mr Coulson to be sacked, comparing his case to that of Damian McBride, the Downing Street aide forced to quit over e-mail smears against senior Conservatives.
Mr Coulson has yet to comment on whether he knew about phone hacking at the paper but the Guardian has said it has no evidence to suggest he did.
The Commons Culture Committee has said it will re-open its inquiry into the affair and may call Mr Coulson and senior executives at News International - the publisher of the News of the World - to give evidence.
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