He became heir apparent on the death of his grandfather King George VI in 1952 when he was only three years old, and is now the longest serving heir to the throne in British history.
"They both agreed there was little they could do but they are very, very annoyed", a source told the publication.
The rumours of Prince Philip's death originated on Twitter during the late hours of Thursday, with the speculation becoming a top trending topic by the end of Friday.
Buckingham Palace was said to have been flooded with calls from concerned members of the public, leaving Her Majesty forced to call her husband for a tense conversation, reports the Sun.
The exhibition opens to the public tomorrow and boasts over 100 personal pieces from the Prince including photos, paintings and other belongings.
'The Queen is absolutely fuming over this and thinks it's disgraceful'.
Buckingham Palace has also commissioned a range of souvenirs for fans of the royal family to purchase - including a gold embellished pram with faux pearl detailing and "Royal Baby 2018" stitched on the back that neatly matches the shoes.
The source added: "If he was ill, she would be with him instead of starting her Scottish holiday".
Below are marble busts of the prince's great-great-great-grandparents Queen Victoria and Prince Albert, who themselves were well-known art lovers.
The Queen's first encounter with Prince Philip was when they were children, at a wedding in 1934, while it is thought that she had been in love with the former Greek royal since she was 13.
Speaking in the audio commentary, Prince Charles says: "his striking cedar wood pavilion stands at over four-and-a-half metres high and features intricate carvings which draw on the rich heritage of Afghan design".