The News of the World's owner has said sorry over the phone hacking scandal and is to set up a compensation fund.
News International says, in some cases, it will apologise and admit liability. The BBC understands it hopes to pay out less than £20m in total to victims including actress Sienna Miller.
A lawyer for some alleged victims said it was a "step in the right direction".
A News of the World reporter and an ex-news editor were arrested and bailed earlier this week over the allegations.
The BBC's business editor Robert Peston called it an "absolutely dramatic development" and said the company believed most claims would be settled for less than £100,000 each.
He said News International was offering to settle with eight people, including well-known names such as former Culture Secretary Tessa Jowell, designer Kelly Hoppen and sports commentator Andy Gray.
In addition to Sienna Miller, the others are believed to David Mills, lawyer and Mrs Jowell's estranged husband; Joan Hammell, former aide to former Deputy PM John 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.
"Past behaviour at the News of the World in relation to voicemail interception is a matter of genuine regret," it said in a statement.
"It is now apparent that our previous inquiries failed to uncover important evidence and we acknowledge our actions were not sufficiently robust."
The firm, which also owns the Times and Sun newspapers, 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, 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, including celebrity publicist Max Clifford, who received a reported £1m.
Mr Clifford said the News of the World's latest offer for other victims was the "right thing to do".
"They have conducted a thorough investigation and police are conducting one which they should have done a few years ago," he said.
There are 24 active cases and they are being heard by Mr Justice Vos.
News International has approached the High Court judge with a way of settling all the cases as a group, and he is planning to hold a case conference for all the active cases next 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".
"Offers will have to be looked at and considered but they should have done this in the first place," he said.
He said the final compensation could run into million of pounds and the individual amounts would vary case by case.
"These individuals have had their privacy invaded. For some, it has had far-reaching effects on their career," he added. "Some people have lost their jobs."
Former Sunday Times editor and broadcaster Andrew Neil said it was a "complete mea culpa".
"They are trying to close it down with their cheque book but I don't think it will succeed," he told the BBC.