To a Danish tabloid, he said the royal staff had shunted him into "third place in the royal hierarchy".
Over recent days, Margrethe and members of the royal family have together or individually visited the prince at Copenhagen's university hospital.
According to PEOPLE, Prince Henrik was first diagnosed with pneumonia while on a trip to Egypt in January.
The palace said he had dementia following his announcement.
Following his wish, he will be cremated, with half his ashes spread over Danish seas and the other half buried in the royal private garden at the Fredensborg Castle.
The royal families in neighbouring Sweden and Denmark sent condolences.
Prince Henrik of Denmark, husband and consort of Queen Magrethe II of Denmark, has passed away.
With a jovial face framed by understated glasses, the prince had a reputation as a bon vivant who enjoyed cooking, art and wine.
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According to Square Enix , there are roughly three worlds per game section, but that doesn't equate to nine total worlds. The early worlds are about 90% complete, with the middle worlds being at about 60%. "It is bound to surprise everyone".
His flamboyant style was both loved and criticised by Danes.
The palace said in a brief statement that the 83-year-old French-born Prince Henrik's condition remains "serious".
He retired from public service in 2016, and the following year he revealed that he did not want to be buried next to his wife because he was never made her equal.
The prince, who retired from public life previous year, was born Henri Marie Jean André de Laborde de Monpezat in a suburb of Bordeaux, France, in 1934.
Before marrying Queen Margrethe, he served in the French Army in the Algerian War and worked for the French Foreign Affairs ministry at the French Embassy in London.
He previously said: "My wife has decided that she would like to be Queen, and I'm very pleased with that".
Henrik was a divisive figure in Denmark and struggled to win the hearts of many throughout his 50-year marriage to the Queen.
By the time Margrethe ascended the throne in 1972, the couple had two young children: Prince Frederik, born in 1968, and Joakim, born in 1969.