According to BBC News, Murdoch will talk in private with Sally and Bob Dowler this afternoon in central London, following claims that the News of the World hacked into Milly's mobile phone after she went missing in 2002.
The meeting also comes after the dramatic resignation of News International chief executive Rebekah Brooks, who was editor of the Sunday tabloid at the time of the alleged hacking.
Mark Lewis, the solicitor representing Milly Dowler's family, told the BBC: "She should have gone ages ago. It happened on her watch. The Dowler family shouldn't have had to wait for this moment. We can't gloat on this position but it's right, it's what should have happened."
Brooks, Rupert Murdoch and James Murdoch have agreed to appear before the commons culture, media and sport committee next Tuesday to answer questions about phone hacking. In a statement today, News International chairman James Murdoch said that the session would give them a chance to show their "determination to put things right".
John Whittingdale, the committee chairman, said that he believed Brooks's resignation had been "inevitable".
"I think many people expected it to come rather sooner, but I think her position was extremely difficult," he said.
"I think the most shocking revelation of all, perhaps, was the hacking of Milly Dowler's phone, which took place when she was the editor of the paper and there has obviously been a stream of revelations since then."
This weekend, News International will run adverts in all of its national newspapers apologising to the nation for the phone hacking scandal. The centrepiece of this will be a letter signed by Murdoch and headed with the words "We are sorry", in giant letters.
The letter reads: "The News of the World was in the business of holding others to account. It failed when it came to itself.
"We are sorry for the serious wrongdoing that occurred. We are deeply sorry for the hurt suffered by the individuals affected. We regret not acting faster to sort things out. I realise that simply apologising is not enough.
"Our business was founded on the idea that a free and open press should be a positive force in society. We need to live up to this.
"In the coming days, as we take further concrete steps to resolve these issues and make amends for the damage they have caused, you will hear more from us. Sincerely, Rupert Murdoch."