Mr Gove, an ardent leave campaigner, was one of just four ministers present at a meeting of the Privy Council held at Windsor Castle in April 2011 when it is thought Nick Clegg had a conversation with the Queen about the union.
The Sun newspaper claims Her Majesty outlined her dissatisfaction at the EU.
A spokesperson for Mr Gove has refused to comment publicly on the claims, however Prime Minister David Cameron appeared to accept that he was not involved.
"These are very serious matters but as far as I can see Michael Gove has made clear that he has no idea where this story came from either," he told BBC Oxford.
Buckingham Palace has lodged a formal complaint with the Independent Press Standards Organisation (Ipso) about the report in The Sun which appeared under the headline "Queen Backs Brexit".
Mr Cameron said: "The Palace has made a very clear statement, the former deputy prime minister has made a very clear statement saying that this didn't happen and I think we should leave it at that.
"There is obviously a proper investigation now being held by the press complaints commission and I think we should let them do their work."
The editor of The Sun, Tony Gallagher, has strongly defended his paper's reporting and the "Queen Backs Brexit" headline - even though she did not explicitly advocate leaving the EU in either of the reported conversations referred to in the story.
"Multiple sources - two sources to be precise - came to us with information about the Queen and her views on the EU and we would have been derelict in our duty if we didn't put them in the paper. It's as simple as that," he told the BBC Radio 4 Today programme.
"You are going to have to take my word for it that we are completely confident that the Queen's views were expressed exactly as we have outlined them both in the headline and the story.
"It is also the case that we knew much more than we published and that remains the case.
"We are in no doubt that the story is accurate."
Labour MP Wes Streeting has written to the Cabinet Secretary Sir Jeremy Heywood urging him to investigate the leak, saying it would be an "extremely serious breach" of Privy Council rules if it came from one of their meetings.