It is understood compensation will be offered to eight people, including the four pictured above The owner of the News of the World hopes to draw a line under the phone-hacking scandal after apologising unreservedly and offering compensation.
News International said it would admit liability in some cases and has put aside about £20m for compensation.
It has admitted there were at least eight victims, whom the BBC understands to include actress Sienna Miller and former Culture Secretary Tessa Jowell.
However, the admission risks opening the company up to more legal action.
For years News International insisted there had been just one “rogue” reporter involved in the hacking of phones.
BBC media correspondent Torin Douglas says now more public figures could sue, MPs claim Parliament has been misled, and some lawyers say there is likely to be greater pressure on the police to bring charges.
Liberal Democrat deputy leader Simon Hughes, whose phone messages were hacked, said there was more than just a financial settlement at stake.
He said: “We need to not just be seeing this as a money pay-out to draw a line and silence people.
“If people have committed serious criminal offences, either those who have already been arrested or others, they need to be pursued through the courts and sent to prison because this is a completely unacceptable practice.”
With News Corporation’s bid for BSkyB still awaiting government approval, former Deputy Prime Minister Lord Prescott said the takeover should be blocked until all the investigations are complete.
He said: “I say this to the government – don’t you dare make this announcement while the Commons and Lords are in recess unless you’ve done a proper judgement on this company.
“And what they’ve said to me is ‘don’t get the plurality of the media mixed up with criminal offences’.”
BBC business editor Robert Peston called News International’s move an “absolutely dramatic development” and said the company believed most claims would be settled for less than £100,000 each.
He said it was offering to settle with eight people, including well-known names such as Ms Jowell, designer Kelly Hoppen and sports broadcaster Andy Gray.
In addition to Ms Miller, the others are believed to David Mills, the lawyer and Mrs Jowell’s estranged husband; Joan Hammell, former aide to Mr Prescott; Nicola Phillips, assistant to publicist Max Clifford; and former Olympian and talent agent Sky Andrew.
News International said the announcement related to voicemail interception between 2004 and 2006, and it followed an “extensive internal investigation” and disclosures through civil cases.
The firm said it had asked its lawyers to “establish a compensation scheme with a view to dealing with justifiable claims fairly and efficiently”.
But it added: “We will, however, continue to contest cases that we believe are without merit or where we are not responsible.”
News International, which is owned by Rupert Murdoch and also owns the Times and Sun newspapers, said it would continue to co-operate with the Metropolitan Police inquiry.
On Tuesday, the News of the World’s chief reporter Neville Thurlbeck, 50, and former news editor Ian Edmondson, 42, were arrested on suspicion of having unlawfully intercepted voicemail messages. They were released on bail until September.
The latest arrests are the first since the Met Police reopened its inquiry – known as Operation Weeting – into claims that staff at the Sunday tabloid had hacked into the phone messages of celebrities and other public figures.
In 2007, the first police investigation led to the convictions and imprisonment of then News of the World royal editor Clive Goodman and private investigator Glenn Mulcaire, who was employed by the paper.
Four alleged victims have reached out-of-court settlements with the newspaper and there are currently 24 active cases being heard by High Court judge Mr Justice Vos.
News International has approached the judge with a way of settling all the cases as a group, and he is planning to hold a case conference on Friday.
Mark Lewis, a solicitor involved in several of the current cases, told the BBC the paper’s apology was a “step in the right direction”.
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