The owner of Britain's News of the World tabloid has offered an "unreserved apology" for phone hacking and said it would set up a compensation fund.
Rupert Murdoch's News International said it admitted liability in some cases brought against the News of the World, Britain's top-selling newspaper.
Actress Sienna Miller is believed to be among those offered a settlement.
The apology came three days after police arrested the Sunday tabloid's chief reporter and a former news editor on suspicion of unlawfully intercepting mobile phone voicemail messages.
News International, which also publishes The Sun, The Times and the Sunday Times, admitted its first internal inquiry into the hacking "failed to uncover important evidence".
"Following an extensive internal investigation and disclosures through civil legal cases, News International has decided to approach some civil litigants with an unreserved apology and an admission of liability in cases meeting specific criteria," the group said in a statement.
It added: "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 then were not sufficiently robust."
The group said it had asked its lawyers to establish a compensation scheme "with a view to dealing with justifiable claims fairly and efficiently".
"This will bring the process of bringing these cases to a fair resolution with damages appropriate to the extent of the intrusion," it said.
"We will, however, continue to contest cases that we believe are without merit or where we are not responsible."
The BBC reported that News International has earmarked about £20 million ($32.5 million, 22.5 million euros) in total for the compensation fund, but that it expected most payments to be about £100,000.
The statement did not say which cases it had decided to settle, but apart from Sienna Miller, former cabinet minister Tessa Jowell and top football agent Sky Andrew were reported to be among others offered payoffs.
Miller, who has an on-off relationship with actor Jude Law, obtained a High Court ruling on Tuesday ordering Vodafone to disclose data relating to other mobile phone users so she can identify who tried to access her voicemails.
In 2007, the initial police investigation into phone hacking led to the convictions and jailing of the News of the World's royal correspondent Clive Goodman and a private investigator.
It emerged they had hacked into the mobile phone messages of Princes William and Harry to obtain information for use in stories.
The News of the World insisted then that Goodman and investigator Glenn Mulcaire were acting alone, but London's Metropolitan Police reopened their investigation earlier this year after fresh revelations suggesting the practice was widespread.
That move prompted the tabloid's former editor, Andy Coulson, to quit as head of communications for Prime Minister David Cameron.
Coulson had resigned as editor when Goodman was jailed but insisted he knew nothing of the phone hacking. He continues to deny wrongdoing, but said the new investigation was distracting him from his job with Cameron.
On Tuesday, police arrested the News of the World's chief reporter, Neville Thurlbeck, and a former news editor, Ian Edmondson. They were bailed to return to a court in September.
Chris Bryant, a lawmaker from the opposition Labour Party who also believes his phone was hacked, called for a judicial review into the case and dismissed the compensation offers.
"There wasn't just one reporter. There weren't very few victims. Nobody ever did a thorough investigation in 2006 or 2009," he said.
"Senior figures at the paper must have been aware of what was going on and should now resign."