Listeners to this morning's edition of Desert Island Discs will hear Mr Cameron talk of how Yorke played a song specially for him at a recent show after requesting the track beforehand. But the left-leaning singer has responded by saying it did not happen. Despite what Mr Cameron thinks, the choice of song was nothing to do with him, Yorke said.
Mr Cameron's latest effort at claiming celebrity kudos comes just a week after he wangled a ticket for the Beckhams' party at which he hobnobbed with stars such as Wayne Rooney, Robbie Williams, David Blaine and Elle MacPherson.
Radiohead - whose hits include Paranoid Android and Karma Police - are one of Mr Cameron's favourite bands, and he was invited along to a Friends of the Earth benefit show being played by Yorke and the band's guitarist, Jonny Greenwood.
Appearing as today's desert island castaway, Mr Cameron tells Sue Lawley that the musicians performed the song "Fake Plastic Trees" at his request. Ms Lawley asks "And he played this song for you?", to which Mr Cameron responds: "Yes, he did. I sent this rather sad letter saying I'd love to come to the concert, thank you for asking. PS please play this, my favourite song - and he did."
But a spokesman for Yorke said yesterday: "The set list was chosen to reflect the fact that it was just Thom and Jonny playing and the choice of songs was what they wanted to play - nothing to do with any special requests."
Yorke is a supporter of many causes such as fair trade, Amnesty International and Friends of the Earth, but has avoided any party alignment. He has been critical of the Labour government in songs such as "You and Whose Army?". He also snubbed a proposed meeting with Mr Blair on environmental issues.
As well as "Fake Plastic Trees", Mr Cameron's Desert Island Discs include The Smiths' "This Charming Man" and "Ernie, the Fastest Milkman in the West" by Benny Hill. The one he would least like to leave behind would be a live version of Bob Dylan's "Tangled Up in Blue".
During his interview Mr Cameron speaks movingly about his disabled son Ivan, who has a rare condition combining epilepsy and cerebral palsy which means that he needs 24-hour care.
Dave's discs: From Dylan to Benny Hill
1 Tangled Up in Blue - Bob Dylan
2 Ernie, the Fastest Milkman in the West - Benny Hill
3 Wish You Were Here - Pink Floyd
4 On Wings of Song (Mendelssohn) - Kiri Te Kanawa
5 Fake Plastic Trees - Radiohead
6 This Charming Man - The Smiths
7 Perfect Circle - REM
8 All These Things that I've Done - The Killers
Luxury item: A case of malt whisky from Jura
Book: 'The River Cottage Cookbook' by Hugh Fearnley-Whittingstall