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.

Russell sustained the injury in Glasgow Warriors' Pro12 semi-final defeat by Connacht in May.
"We don't expect him to play in the first two or three games of the season," Townsend told BBC Scotland.
"We'll just see what happens over the next few weeks."
Russell was stretchered off during Glasgow's defeat in Galway after a sickening collision with team-mate Zander Fagerson in the first minute of the match.
The Scotland playmaker was kept in hospital for over a week for observation, but is now back in light training.
"He's working on skills, he's very keen to be involved," Townsend continued.
"Everything has gone well so far with his recovery, but obviously we're being very cautious on when he can come back to training with the team and also full training.
"So far his progress has been really good."

That includes broadband operators, schools, universities and libraries. Ofcom said that the savings could be passed on to consumers.
It follows a review that found the former state monopoly had "significant" market power.
BT said that more regulation could damage investment in infrastructure.
"These are proposals for discussion, so we'll be making our views known to Ofcom. We don't expect a final decision for some time," the company said in a statement.
"We believe there should be less regulation in this market, not more, as businesses already have diverse and growing choice amongst a large number of providers."
The consultation - which runs until July - will consider a form of charge control that aims to bring prices down over a three-year period.
The charge controls relate to two groups of services - older leased lines that offer speeds up to 8Mbps (megabits per second) and newer Ethernet lines that offer speeds up to 1Gbps (gigabit per second).
"The slow speed traditional service on up to 8Mbps leased lines is not likely to help mobile and other broadband operators, since slow lines like this are not sufficient for modern backhaul," said Andrew Ferguson, editor of broadband news site ThinkBroadband.
"The real market is the Ethernet sector where the price reduction will be very welcome by those buying these lines."
But he added: "There is also the real risk that BT - if revenue is reduced in one area -  will seek to maintain profits by increasing prices elsewhere."
Ofcom is also proposing that companies providing leased lines should be granted access to BT's networks through a process known as dark fibre.
This would involve BT giving competitors physical access to its fibre-optic cables, allowing them to take direct control of the connection.
It is called dark fibre because the cable would not be lit using BT's electronic equipment. Instead the competitor would install its own equipment at either end of the cable.
Ofcom expects to publish its decision in the first quarter of next year, with any price cuts coming into effect in April 2016.
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.
[NEXT_CONCEPT]
Glasgow Warriors head coach Gregor Townsend says Finn Russell is progressing well in his recovery from a serious head injury, but still expects him to miss the start of the season.
[NEXT_CONCEPT]
UK regulator Ofcom is proposing cutting what BT charges for its leased telecoms lines, which could see significant price cuts for those who buy them.