News International says it has approached some claimants with an "unreserved apology".
It will also establish a compensation fund, with a view to "dealing with justifiable claims efficiently".
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.
"I understand the company's hope would be that in total it will pay out less than £20m," he added.'Genuine regret'
There are 24 active cases and they include claims of breach of privacy brought by the film star Sienna Miller, former culture secretary Tessa Jowell, football commentator Andy Gray and the designer Kelly Hoppen.
News International, which also owns the Times and Sun newspapers, said: "Past behaviour at the News of the World in relation to voicemail interception is a matter of genuine regret.
"It is now apparent that our previous inquiries failed to uncover important evidence and we acknowledge our actions were not sufficiently robust."
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. There 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 into phone hacking 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.
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