The paper claims Number 10 is using the Lascelles Principles, a set of 70-year-old rules that state the monarch must follow her Prime Minister’s advice.
A source said: “Boris won’t resign even if he loses a no-confidence vote, and it is not within the sovereign’s constitutional powers to make him.”
They added: “The PM will advise the Queen of that and she must follow her Prime Minister’s advice. That’s how this country works.”
The move would risk bringing the Queen, who traditionally remains politically neutral, deeper into the Brexit crisis.
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However, there is uncertainty over whether Mr Johnson could actually enact the Lascelles Principles, with some arguing that they are only designed to allow a monarch to refuse to call a general election.
Others suggest the passing of the Fixed-Term Parliaments Act in 2011 meant the principles were essentially abandoned and that the Queen is still within her rights to sack a PM if they do not enjoy the confidence of the Commons.
Labour peer and barrister Lord Falconer tweeted of the plan: “One of the conditions in the Lascelles Principles for refusing a dissolution was [that the] Monarch could rely on finding another Prime Minister who could carry on Government, for a reasonable period, with a working majority in the House of Commons.”
He added: “If there is no alternative PM who could command a majority after Boris Johnson loses a vote of no confidence then there will be nobody for who HMQ can send to form a government.
“But if there is such an alternative, then Boris Johnson will have to vacate Downing Street.”
Royal historian Kate Williams also dismissed the plans, tweeting that they were “utter nonsense”. She described the briefing as “a propaganda war by No 10 against the Palace – and a threat to the Queen that they will spin against her, so she should fall into line”.
Opposition MPs have still yet to call a no-confidence vote against Mr Johnson, fearing they do not have the numbers – or a credible caretaker PM to take over.
Lib Dem leader Jo Swinson has said she could not support Labour’s Jeremy Corbyn becoming temporary Prime Minister.