The person said the company, Rupert Murdoch’s News International, had offered about $3.2 million to the Dowler family, with an additional payment of about $1.6 million to go to charity.
The News of the World is believed to have hacked into the voice mails of hundreds, possibly thousands, of people, including celebrities and politicians.
Widespread public revulsion resulted from the revelation in July that its victims included the 13-year-old Milly Dowler, who disappeared in 2002 and whose messages were intercepted and erased to make room for new ones even as her family searched for her.
That disclosure ultimately led the Murdochs to close The News of the World, the government to open several inquiries into the matter and the police to expand their criminal investigation into the paper’s conduct.
Mr. Murdoch, who seemed unperturbed by earlier accusations about his employees’ phone hacking, was demonstrably shaken by the news about Milly Dowler. He met with the Dowler family to apologize, and later told a parliamentary committee that he had been humbled by the episode.
News International would not disclose the amounts currently under discussion.
If both parties agree to the proposal, the settlement will be by far the largest to date in the hacking case. In 2008, Gordon Taylor, a soccer executive, received about $1.4 million from News International, the British newspaper arm of Mr. Murdoch’s News Corp.