In a piece for the New Statesman, Grant explained how he arranged a meeting with Paul McMullan, who made several claims about the ongoing phone hacking scandal.
According to Grant's transcript, McMullan said that Downing Street's former director of communication Andy Coulson "knew all about" phone hacking of celebrities by News of the World staff when he was editor of the Sunday tabloid.
"He wanted to cover his tracks all the time," McMullan allegedly said. "So he wouldn't just write a story about a celeb who'd done something. He'd want to make sure they could never sue, so he wanted us to hear the celeb like you on tape."
McMullan also claimed that [Prime Minister David] Cameron "must have known" about the phone hacking. "That's the bigger scandal," he continued. "Cameron's either a liar or an idiot."
However, when Grant asked McMullan if News International boss Rupert Murdoch was aware that reporters were listening to private voicemails, he replied: "Possibly not. He's a funny bloke given that he owns The Sun and the Screws (News of the World)... quite puritanical... I don't even think he really worried himself too much about it."
Towards the end of the transcript, when asked if he had ever listened to Grant's calls, McMullan said: "I didn't personally ever listen to your voice messages. I did quite a lot of stories on you, though. You were a very good earner at times."
Earlier this week, the News of the World printed a public apology for the scandal.
Celebrities including Sienna Miller are currently involved in civil actions against the paper in relation to claims that their voicemails were listened to by its reporters.