John Prescott led Labour calls for Mr Coulson to be fired after it was reported that News Group, which owns his former Sunday paper, spent more than £1 million on out-of-court settlements relating to the claims.
But the Conservative leader indicated he had no intention of losing his director of communications, who resigned from the News of the World in 2007 after one of its reporters was jailed for hacking into the phone messages of royal aides.
The latest claims, in The Guardian, suggest that the practice was much wider than previously thought.
Speaking outside his west London home this morning, Mr Cameron said: "It's wrong for newspapers to breach people's privacy with no justification.
"That is why Andy Coulson resigned as editor of the News of the World two-and-a-half years ago.
"Of course I knew about that resignation before offering him the job. But I believe in giving people a second chance.
"As director of communications for the Conservatives, he does an excellent job in a proper, upright way at all times."
MPs from all three parties, including former deputy prime minister Mr Prescott and Cabinet minister Tessa Jowell were among the targets of the alleged phone taps.
The Guardian quoted sources saying police officers found evidence of News Group staff using private investigators who hacked into "thousands" of mobile phones.
Mr Prescott said: "If these allegations are to be believed, the enormity of it is unbelievable.
"I am actually staggered that Mr Cameron, who employed Mr Coulson, who was the editor at the time through all these allegations, says he's quite relaxed about these allegations."
Former home secretary Charles Clarke described the alleged practices as "outrageous".
"I think that David Cameron has to sack Andy Coulson because his denial is very narrow in the extreme," he said.
"I think David Cameron himself has to be much clearer about the situation.
"I think that the Home Secretary should be asking the Chief Inspector of Constabulary for a full report about the police behaviour in this whole incident."
PR agent Max Clifford, who is another whose phone was allegedly hacked into, said the claims raised "lots of serious questions".
Mr Clifford, who works with some of Britain's best known celebrities, said: "If these allegations prove to be true, then it's something that an awful lot of people are going to be very unhappy about."
An influential Commons committee is poised to summon newspaper chiefs. It is holding an emergency meeting today.
Tory MP John Whittingdale, who chairs the Culture, Media and Sport Select Committee, said he wanted to reopen an inquiry into the issue.
Mr Whittingdale said: "My view is that this has raised very serious questions about the evidence given to us.
"There are a number of questions I would like to put to News International on the basis of what The Guardian has reported."
The committee would examine the circumstances "as a matter of urgency" at a scheduled meeting later today, he said.
"It may well be that we decide we wish to have somebody from News International to appear before us."
He said he had seen no "direct evidence" that assurances previously given to the committee by the publisher on the matter had been untrue.
But he added: "If that is the case, it does beg the question why News International have apparently paid huge sums of money in settlement of actions in the courts.
"That is a question I would wish to put to News International."
The Guardian said Andy Coulson was deputy editor and then editor of the News of the World when journalists were using the private investigators.
Mr Coulson resigned from the News of the World after royal editor Clive Goodman was sentenced to four months in prison in January 2007 for plotting to hack into telephone messages belonging to royal aides.
The Guardian said the £1 million paid out by News Group to secure secrecy concerned three out-of-court settlements in cases that would have shown the alleged methods being used.
One of the settlements, totalling £700,000 in legal costs and damages, involved legal action brought by Gordon Taylor, chief executive of the Professional Footballers Association, the newspaper said.
In the Goodman trial, Mr Taylor was revealed as one of the public figures whose phone messages were illegally intercepted by private investigator Glenn Mulcaire.
Mr Coulson said last night: "This story relates to an alleged payment made after I left the News of the World two-and-a-half years ago.
"I have no knowledge whatsoever of any settlement with Gordon Taylor.
"The Mulcaire case was investigated thoroughly by the police and by the Press Complaints Commission. I took full responsibility at the time for what happened on my watch but without my knowledge and resigned."
Liberal Democrat home affairs spokesman Chris Huhne compared the affair to the Damian McBride smears scandal.
Mr McBride was forced to resign as Gordon Brown's director of strategy and communications after it emerged he had written emails suggesting possible smears of senior Tories.
"At the very least, Andy Coulson was responsible for a newspaper that was out of control and at worst he was personally implicated," Mr Huhne said.
"Either way, a future prime minister cannot have someone who is involved in these sort of underhand tactics. The exact parallel is with Damian McBride."
The Metropolitan Police Service (MPS) said in a statement: "The MPS carried out an investigation into the alleged unlawful interception of telephone calls.
"Officers liaised closely with the Crown Prosecution Service. Two people were charged and subsequently convicted and jailed. We are not prepared to comment further."
A spokeswoman for News International, the parent company of News Group Newspapers, said: "News International feels it is inappropriate to comment at this time."
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