According to The Guardian the papers' owners News International was forced to pay more than £1m in secrecy payments and out of court settlements to individuals including Professional Football Players union boss Gordon Taylor.
The News of the World and The Sun's editors at the time of the wrongdoing were respectively the current Conservative Party director of communications Andy Coulson and Rebecca Wade, who is due to assume the role of News International CEO.
The settlements were made after following the disclosure that its royal editor Clive Goodman had accessed voicemails of Buckingham Palace staff.
In one month alone The Guardian is alleging the papers targeted the mobiles of personalities and politicians including Gwyneth Paltrow, John Prescott, George Michael, Boris Johnson, Tessa Jowell (who was responsible for media as culture secretary) and Jade Goody.
Others named include Sven-Goran Erickson, Elle Macpherson, Max Clifford, Jeffrey Archer, Nigella Lawson and Lenny Henry.
The Guardian story appears to be based on information from two sources, one based at the Metropolitan Police, who had access to police evidence from the investigation.
One of the sources said private investigators had hacked into "two or three thousand" mobiles.
Both The Times and Sky News coverage of the allegations have focused on the involvement of Cameron's communications chief Coulson and whether he is fit to remain in the post.
Cameron has since confirmed his position will be unaffected.
Other accusations connected to the settlement with Taylor include the revelation that the Information Commissioner has evidence concerning the employment by newspapers of a private investigator who specialised in hacking into police computers.
News International claimed at the time of Goodman's jailing in January 2007 that his actions were an exception.
It was unable to comment on The Guardian's allegations.
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