The 83-year-old has long complained at being named as prince rather than king.
It had been expected that the pair would be buried together in a specially-designed sarcophagus at Roskilde Cathedral.
The queen, 77, is said to have accepted her husband's decision.
"It is no secret that the prince for many years has been unhappy with his role and the title he has been awarded in the Danish monarchy," the Royal Danish House's director of communications told the tabloid BT.
"For the prince, the decision not to be buried beside the queen is the natural consequence of not having been treated equally to his spouse - by not having the title and role he has desired."
Born Henri Marie Jean AndrÃ© de Laborde de Monpezat in France in 1934, he and the queen have two sons. Henrik retired from public life last year.
The royal house said despite rumours to the contrary the prince wants to be buried in Denmark rather than France.
Prince Henrik of Denmark will not be buried next to his wife Queen Margrethe, royal officials say, with the prince unhappy at never being designated her equal.