Danish Prime Minister Helle Thorning-Schmidt took a photograph of herself with US President Barack Obama and Cameron at Mandela's memorial service last week, which many thought was in bad taste.
Thorning-Schmidt was so embarrassed by the furore that she secretly pledged to destroy the image but Cameron has pleaded with her to auction it for charity, a Sunday Times report said quoting sources.
During a private conversation last week, Cameron suggested to Thorning-Schmidt that it could raise a substantial sum, the report said.
Critics have accused Cameron of debasing the office of prime minister and showing disrespect for the solemnity of the memorial service where the picture was taken.
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He has since made light of the episode, saying he was just being polite when he agreed to Thorning-Schmidt's request.
The Danish press appears to have been more forgiving of Thorning-Schmidt - whose father-in-law is the former Labour leader Neil Kinnock - than the British media have been of Cameron, the report noted.
A Downing Street spokesman did not deny the conversation between Cameron and Thorning-Schmidt had taken place but refused to elaborate, it said.
The so-far unseen selfie would be likely to spark a fierce bidding war.