Long vocal about his frustration over being relegated to a supporting role, the 83-year-old prince said he was not on equal footing with his wife in life and therefore did not want to be so in death.
That decision has been accepted by the queen, the palace's communications chief Lene Balleby told Danish daily BT.
A second palace spokesman confirmed the accuracy of the newspaper report to AFP but provided no other details.
Disappointed that his royal title of prince consort was never changed to king when his wife became queen in 1972, Henrik has often spoken out about his discontent, which did little to endear him to his subjects.
The couple will break with royal tradition by not being buried together in the Roskilde Cathedral.
The prince consort nonetheless intends to be buried in Denmark.
He "loves Denmark and worked for Denmark for more than 50 years. The prince therefore wants to be buried in Denmark, but the details have not been settled yet," Balleby said.
Born Henri Marie Jean Andre Count de Laborde de Monpezat on June 11, 1934 in Talence, near Bordeaux, he met Margrethe, then the crown princess, while he was stationed in London as a diplomat.
Upon marrying her, he changed his name to Henrik, converted from Catholicism to Protestantism and renounced his French citizenship to become a Dane.
By the time Margrethe acceded to the throne, the couple had two young children: Prince Frederik, born in 1968, and Joakim, born in 1969.
Henrik retired from public service in January 2016.