The victims, who are suing the paper, include actress Sienna Miller and her ex-husband David Mills, former Culture Secretary Tessa Jowell and former Sky Sports commentator Andy Gray, The Guardian reported.
News International said its previous investigations of the scandal that has shaken British politics and media were "not sufficiently robust" and issued an "unreserved apology." It is offering to pay unspecified damages and legal fees.
Altogether, 24 people have begun legal action, but the company said not all claimants had presented sufficient evidence.
No executives are expected to lose their jobs.
News International had blamed the hacking on "rogue reporter" Clive Goodman. He was jailed in January 2007 along with private investigator Glenn Mulcaire for intercepting voice mail messages to members of the royal household.
Andrew Neil, a former Murdoch executive, told the BBC: "This is one of the most embarrassing apologies I've ever seen from a major British corporation."
He predicted that settling civil suits would not affect the current criminal investigation.