The Queen is reported to have asked dinner guests to give her “three good reasons why Britain should be part of Europe”, according to royal author Robert Lacey.
Mr Lacey, writing for The Daily Beast website, said the monarch posed the challenge recently.
“'Give me three good reasons', she has apparently been asking her dinner companions recently, 'why Britain should be part of Europe?'” Mr Lacey wrote.
A Buckingham Palace spokesman said: “The Queen is above politics and acts on the advice of her Government in political matters.
“The referendum is a matter for the British people to decide.”
Mr Lacey told the Press Association: “The Queen likes a healthy debate around the dinner table. It was just a question.
“She's aware of the complexities for different parts of the UK.
“As we know, she's very careful not to betray whatever her personal opinions may be on this. You can say the same of her husband.”
The Queen's grandson Prince Harry broached the subject of the EU referendum at the street party for the monarch's official 90th birthday.
He brought up the debate over Brexit with dairy farmer Mike King, 46, and his wife Rachel, 48, from Iron Acton, near Bristol, as he chatted to them on The Mall.
But the Prince kept his own views close to his chest. Mr King said afterwards: “I told Harry we were farmers and were hardy so we're just used to the wet (weather).
“When he heard we were farmers, he said 'In or Out?' and I said I was going to ask him the same.
“I said 'I'm undecided', and he said: 'I'm not allowed to say'.”
Traditionally, the Queen and her family never vote or stand for election.
Although the law does not ban royalty from voting, it is considered unconstitutional for them to do so.
Last month, the press watchdog ruled that The Sun newspaper had breached press regulations with its front page headline suggesting the Queen was in favour of the UK leaving the EU.
The headline, “Queen backs Brexit”, published in March, was “significantly misleading”, the Independent Press Standards Organisation (Ipso) said.
Buckingham Palace complained, insisting the Queen was “politically neutral”.
Ahead of the Scottish independence referendum in September 2014, the Queen told a well-wisher in the crowd after morning church at Crathie Kirk, near Balmoral, a few days before the vote: “Well, I hope people will think very carefully about the future.”
When Scotland voted no, Prime Minister David Cameron confirmed the Queen's delight at the decision when he was caught on camera saying: “She purred down the line.”
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