Sources said Mr Trump has requested a trip to the Cabinet War Rooms, where Mr Churchill planned military strategies during World War 2.
The Republican is believed to have been inspired by the blockbuster film Darkest Hour, which stars British actor Gary Oldman as the wartime PM.
Mr Trump recommended the flick to Theresa May when the two leaders spoke on the phone in December, according to a White House insider.
The President’s state visit, which has been delayed several times, is due to take place between October 5 and October 8.
A source close to Donald Trump‘s administration told the Sunday Times: “He wants to visit the Churchill war rooms and is very inspired by Darkest Hour.”
A senior Whitehall source insisted plans for the state visit were at a preliminary stage.
But they added: “It is plausible that a visit to the Cabinet War Rooms will be included in the itinerary.”
Mr Trump has long admired Mr Churchill, whose bulldog spirit during the Second World War inspired the allied forces to defeat Adolf Hitler.
One of his first acts as president was to return a bust of the former Conservative PM to the White House’s Oval Office.
Under Barack Obama’s presidency, Mr Churchill’s bust was sent back to the British embassy and replaced by one of Martin Luther King Jr.
However, the bust now on show is not the one loaned by Tony Blair to George W Bush but one the White House has had since the 1960s.
During Mr Obama’s time in office, it had been displayed in the building’s private residence, rather than the Oval Office.
Republican politician Mike Huckabee compared Mr Trump to Mr Churchill in a tweet after seeing Darkest Hour over Christmas.
He wrote: “Churchill was hated by his own party, opposition party, and press. Feared by King as reckless, and despised for his bluntness.
“But unlike Neville Chamberlain, he didn’t retreat. We had a Chamberlain for 8 years; in Donald Trump we have a Churchill.”
The Cabinet War Rooms, in Westminster, were first opened to the public by Margaret Thatcher in 1984.