It would echo a similar move made in 1957, when the Queen let it be known that both Prince Charles and Princess Anne had been given the polio vaccine in order to counter public fears.
George Eustice, the Environment Secretary, said it was crucial for people to be encouraged to take the vaccine.
He told Times Radio: "This would be a personal decision for the Queen as it is for everyone but it's very important that we try to make sure people are reassured about this vaccine."
Discussions are also said to be under way between Buckingham Palace and Whitehall about a potential role for the Prince of Wales and the Duke of Cambridge in publicising the national vaccination programme.
It follows reports the NHS wants 'sensible' celebrities and the Royal Family to promote the jab to combat scepticism over vaccination.
However the Duke will not be eligible for early vaccination because he is under 50 and has already had coronavirus, so will already carry some immunity. The Prince of Wales also caught a mild dose of the disease in March.
At the time he expressed sympathy with those who had lost family or friends saying he was 'so determined to find a way out of this' to prevent more people from losing loved ones.
In a speech to the nation in April, the Queen also appeared to praise scientists racing to find vaccines and drugs.
"This time we join with all nations across the globe in a common endeavour, using the great advances of science and our instinctive compassion to heal," she said in the televised broadcast.
"We will succeed - and that success will belong to every one of us."
An endorsement from the Royal Family would be seen as a powerful message to counter anti-vaccination misinformation circulating online. There are fears that anti-vaxx messaging could prevent Britain from reaching herd immunity.
But royal aides are wary of being seen to politicise the family and have stressed that any medical decision is a private matter.
A Buckingham Palace spokesman said: "Medical decisions are personal and this is not something we will comment on."
The Royal Family has traditionally held progressive views on vaccination. In 1721, Queen Caroline, wife to George II, when still Princess of Wales, had her own children inoculated against smallpox after learning hearing from her friend, Lady Mary Wortley Montagu, that children in Turkey were being successfully vaccinated.
In his book Letters on England, Voltaire wrote that at least 10,000 children owed their lives to Queen Caroline and Lady Mary.
Queen Charlotte, wife of George III, was also a notable champion of vaccination, despite losing two sons to smallpox.
