The Guardian newspaper has claimed that Mulcaire intercepted messages left by relatives for the 13-year-old in 2002 on behalf of the tabloid. Dowler's family discovered that messages had been cleared, leading to false hopes that she was still alive.
Speaking on BBC Radio 4, Grant recalled how he had bugged ex-News of the World journalist Paul McMullan to discuss the phone hacking scandal. In a transcript published by the New Statesman, Grant asked McMullan if it was true that John Yates, acting deputy commissioner of the Metropolitan Police, had been asked about the possible hacking by a parliamentary committee.
According to the transcript, McMullan responded: "Yeah. Yeah. It's more than likely. Yeah... It was quite routine. Yeah - friends and family is something that's not as easy to justify as the other things."
Grant told presenter Victoria Derbyshire today: "It was pretty revealing and one of the things he did talk about, was that he was pretty sure that the family and friends of Milly Dowler, and in fact the girls murdered at Soham, had been hacked by the News of the World."
Offering his thoughts on the alleged hacking, Grant said that "like everyone" he found it to be "beyond repulsive".
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"The part that [is] particularly scary is not just that this was a national newspaper that [was] doing this, but it's the ramifications, it's the parts of this story that are yet fully to come out, which are; how many newspapers were using phone hacking on a widespread and industrial basis, because it was not, I can assure you, just the News of the World.
The actor continued: "To me, even more terrifying is the fact that successive governments have winked at all this, because they are completely in hock to the tabloids, especially the Murdoch press, to get themselves elected."
Grant said that all political parties are "absolutely terrified" of the company and stated: "If you've never been in the crosshairs of a red-top tabloid, or even a non-red-top tabloid - just a tabloid, you don't know how ruthless they can be, and how terrifying."
When McMullan was brought on air, Grant remained to discuss the story, frequently responding to his comments, stating that "exposing what Jordan's up to, is hardly exposing the Watergate scandal".
Grant concluded: "I'm quite sure Glenn Mulcaire was told specifically, 'Hack Milly Dowler's phone'."
Told that he had no evidence, Grant said that his claim was based on speaking to McMullan "off the record".
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Watch News International corporate affairs' executive Simon Greenberg speak to Sky News this afternoon below: