Milly Dowler's family have been made a £3m offer by Rupert Murdoch's News International in an attempt to settle the phone-hacking case that led to the closure of the News of the World and the resignation of the company's chief executive, Rebekah Brooks.

The money on the table is understood to include a personal £1m donation to charity by Murdoch himself as well as contributions to the family's legal costs. But the publisher has not yet reached final agreement with the Dowler family, whose lawyers were thought to be seeking a settlement figure closer to £3.5m.

The seven-figure sums under negotiation are far larger than other phone-hacking settlements reached – and amount to one of the largest payouts ever made by a newspaper owner – reflecting the fact that the phone-hacking case affected a family who were victims of crime.

Milly Dowler went missing aged 13 in March 2002 and was later found murdered.

The terms of any final settlement are not expected to be confidential. It is less clear, however, whether more detail will emerge about how and when the phone was targeted. The family and their lawyers declined to comment on Monday.

The hacking of Milly Dowler's mobile phone after her death emerged in July. Voicemails were accessed on behalf of the News of the World and messages left for her were deleted to make room for more recordings. This gave the family false hope that she was still alive [see footnote].

On Monday afternoon there was growing speculation that a deal was close, with some involved in the negotiations suggesting a deal could come as soon as this week. However, other sources familiar with the negotiations indicated there were still enough matters unresolved to mean a final settlement would be delayed further.

The actor Sienna Miller accepted £100,000 from News International after the publisher accepted unconditional liability for her phone hacking and other privacy and harassment claims in May. A month later, football pundit Andy Gray accepted £20,000 plus undisclosed costs.

Other lawyers bringing phone-hacking cases have privately indicated that they would be advising many of those bringing actions to try to reach a settlement rather than take their cases to lengthy and expensive trials. A handful of cases have been taken forward as lead actions by Mr Justice Vos, to establish a benchmark for settlements in future lawsuits. However, with the amount of damages alone offered to the Dowler family expected to amount to well over £1m, the settlement easily exceeds other high-profile payout made by newspapers by way of apology.

In 2008, Kate and Gerry McCann, the parents of the missing Madeleine McCann, accepted £550,000 in damages over more than 100 "seriously defamatory" articles published by Richard Desmond's Express newspapers.

This year, eight newspapers paid an unspecified six-figure sum to Chris Jefferies, the landlord of the murdered Joanna Yeates over allegations made against him over the her death. The titles made public apologies to him. Another man, Vincent Tabak, has been charged with her murder, with a trial due next month.

Rupert Murdoch personally met the Dowler family in July, shortly after the story about hacking into her phone broke, making what the family's lawyer, Mark Lewis, said was a "full and humble" apology. The News Corporation chairman and chief executive "held his head in his hands" and repeatedly told the family he was "very, very sorry".

On Monday night, News International confimed it was "in advanced negotiations with the Dowler family regarding their compensation settlement. No final agreement has yet been reached, but we hope to conclude the discussions as quickly as possible."

Sources close to News International said the publisher had initiated the offer of compensation, although at a level lower than the £3m settlement.

• The following was published on 12 December 2011 in the corrections and clarifications column: An article about the investigation into the abduction and death of Milly Dowler (News of the World hacked Milly Dowler's phone during police hunt, 5 July, page 1) stated that voicemail "messages were deleted by [NoW] journalists in the first few days after Milly's disappearance in order to free up space for more messages. As a result friends and relatives of Milly concluded wrongly that she might still be alive." Since this story was published new evidence – as reported in the Guardian of 10 December – has led the Metropolitan police to believe that this was unlikely to have been correct and that while the News of the World hacked Milly Dowler's phone the newspaper is unlikely to have been responsible for the deletion of a set of voicemails from the phone that caused her parents to have false hopes that she was alive, according to a Metropolitan police statement made to the Leveson inquiry on 12 December.