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

It is understood that News International has made a settlement offer estimated by sources at close to £3m, a figure that would include a £1m donation to charity. But the publisher has not yet reached agreement with the Dowler family, whose lawyers were thought to be seeking a settlement figure of closer to £3.5m.

The seven-figure sums under negotiation are far larger than other phone-hacking settlements reached, reflecting the fact that the phone-hacking case affected a family who were victims of crime. Thirteen-year-old Dowler went missing in March 2002 and was later found murdered.

It emerged in July that Milly Dowler's mobile phone had been hacked after her death. 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, because messages were disappearing [see footnote].

On Monday afternoon there was growing speculation that a deal is close, although other sources familiar with the negotiations indicated that there are still enough matters unresolved to mean that an agreement in principle had not yet been reached behind the scenes.

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 Andy Gray accepted £20,000 in damages plus undisclosed costs.

Other lawyers bringing phone-hacking cases are privately indicated that they would be advising many of those bringing actions to try and 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.

Murdoch met with the Dowler family in July, shortly after the original 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".

• 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.

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