The Independent understands that peers are pushing for the US president to be given the opportunity to make a speech in the Royal Gallery, a room outside of Mr Bercow’s direct control.
But it would also be a challenge to Mr Bercow’s authority, at a moment when the commons speaker is already under intense pressure over disputed bullying allegations.
Any invitation to the president will certainly prove controversial with the British public, with campaigners having promised mass protests whenever the US leader comes.
The plan to bring Mr Trump to Westminster has arisen because while the government was planning a state visit last year, Mr Bercow unilaterally vetoed any event involving the president in Westminster Hall – a part of the Palace of Westminster he has some influence over. Barack Obama made a speech there in 2011.
But with Mr Trump’s July visit now pending, peers have launched a drive to open parliament to the president. Lord Cormack told The Independent: “He is the President of the United States, our most important ally.
“My own personal views on him are completely irrelevant – we should give him the opportunity to speak to both Houses.”
He added: “He should be received with proper good manners, and as far as a speech is concerned ... he would speak where most other presidents have spoken – in the Royal Gallery.”
While any event in Westminster Hall must be approved by the room’s three “keyholders” – the commons speaker, the lord speaker and the lord great chamberlain – Mr Bercow has little sway over events in the Royal Gallery.
He is known to have irritated the other key figures in parliament’s hierarchy by vetoing Westminster Hall’s use without first consulting others.
Lord Jones said: “I think [bringing the president to parliament] would be an excellent idea. Trump is the democratically elected leader of our largest trading partner, our biggest inward investor and the guarantor, when all’s said and done, of our security.
“The bosses of China and Saudi Arabia are happily invited.”
Mr Trump recently confirmed he will travel to the UK for a working visit on Friday 13 July, coinciding with a trip to Europe for a Nato summit.
It is possible that the US President could speak in the Royal Gallery at that point, though a major speech is more likely to take place on the full state visit being planned by Downing Street, for which a date is yet to be scheduled.
Number 10 would not comment about the potential for a visit to parliament in July, but sources said arrangements for the visit are on-going.
Another Conservative peer confirmed to The Independent that using the Royal Gallery would be possible, and argued that Mr Bercow’s decision to block the use of Westminster Hall had “interfered with the diplomatic interests of the country”.
The peer added: “You have to accept that in the democratic world you do business with whoever is elected, and in this instance, in relations with the US, Britain has let France steal a march on it.
“Given the way the world is evolving we need to be friendly with the US.
“I think that the adverse reaction to Trump on a personal level is understandable, but on the diplomatic level it is extremely foolish.”
Former cabinet minister Lord Tebbit said he would like to see the lord speaker offer to host the US president for a speech in the upper house and argued there would be support from “a lot of peers for doing so”.
He added: “John Bercow took a unilateral decision. I am not sure he has the power to. We can normally rely on him doing anything that is against the Conservative party.”
Parliamentary sources are certain the lord speaker could offer the use of the Royal Gallery without getting approval from Mr Bercow first.
The commons speaker also appeared to indicate this in the statement he made to MPs as he blocked use of Westminster Hall in 2017, when he said: “So far as the Royal Gallery is concerned ... I do not perhaps have as strong a say in that matter.”
The lord speaker, Lord Fowler, rebuffed Mr Bercow the following day, indicating he is prepared to consider proposals for Mr Trump to speak.
He said: “My view is that I will keep an open mind and consider any request for Mr Trump to address parliament if and when it is made.”
Currently the Royal Gallery is being used to house paintings that are being restored. But there is not thought to be any obstacle to its use by the time of any state visit.
It is lined with the portraits of kings and queens and their consorts, while the doorways and the bay window are flanked by gilded statues of Richard I and Edward III. There are two large wall paintings depicting decisive moments in British history – victory at Waterloo and Trafalgar.
In 1982, Mr Reagan was watched by Margaret Thatcher and other parliamentarians as he predicted the eventual downfall of Communism, famously stating that it would be left on the “ash heap of history”.
King Juan Carlos of Spain, German chancellor Angela Merkel, Mr Clinton, Haile Selassie, Nikita Khrushchev, and François Mitterrand are among others to have spoken there.
Social media is already buzzing with talk of UK anti-Trump protests to coincide with the July trip, with campaigners having promised a “march of millions” if the US president comes.
But the move to allow Mr Trump to attend parliament comes at a critical moment for the commons speaker, who is facing calls to quit following fresh bullying claims – denied by Mr Bercow – from a former private secretary.
Angus Sinclair has said Mr Bercow undermined him by mimicking him, swearing and shouting, and once even smashed a mobile phone in front of him.
David Leakey, who retired from his powerful parliamentary job of being Black Rod, said Mr Bercow was “bringing parliament into disrepute” and called for a full probe into his behaviour.
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