The King and Queen have begun wrapping up their four-day state visit to the US with ‌a short stop at the White House to bid farewell to President Donald Trump.

The official reason for the royal trip was to mark the 250th anniversary of the US winning its independence from British colonial ‌rule, cueing multiple wry jokes from the King in speeches to Washington DC’s elite about being on the losing side of the American Revolutionary War.

But it was also designed to mend what the monarch called in Tuesday’s state dinner with Trump an “unbreakable bond” and “indispensable alliance” between the two countries, lately strained because Britain and other European allies have declined to join the two-month-old US-Israeli war against Iran.

It seemed to work.

As enraged as he has ‌been by British Prime Minister ‌Keir Starmer, Trump has spoken at length about how fond he is of his “great friend” the King, the day after their ​dinner.

“When you like the king of a country so much, it probably helps your relationship with the prime minister,” he said.

On Thursday morning (US time), as the royals, Trump and wife Melania posed for photographs on a red carpet outside the White House’s South Portico, Trump — frequently denounced by domestic political opponents as a would-be king — pointed to the monarch and said: “He’s the greatest king, in my book.”

Donald Trump declares the King as ‘the greatest’ at their farewell. Photo: AAP

The group then went inside only to come back out after just five minutes. The King and Queen then got in their car to tour several sites in ‌Virginia.

“Great people,” ​Trump said toward the departing motorcade.

“We need more people like that in our country.”

During the royal ‌trip, the King has drawn smiles from everyone from lawmakers in the US Congress to Harlem school children at an urban farm in New ​York City.

Among the biggest smiles of all came from Trump himself, as the King revealed a gift for the President at Tuesday’s White House reception: The original bell from the conning tower of a Royal Navy submarine launched from a British shipyard ​in ​1944 and named HMS Trump.

For his final day, the King laid a wreath at Arlington National Cemetery, across the ‌Potomac River in Virginia, a sacred site for many where tens of thousands of the country’s war dead are buried.

On Wednesday the King and Queen commemorated victims of the September 11, 2001 attack on New York City, laying a floral bouquet at the memorial where the World Trade Centre’s twin towers once stood.

The royal couple are also expected to attend a small-town block party in Virginia to join in ​what the British embassy called the “North American tradition” of “a ‘potluck’ meal”.

Later in the day, the King will fly to Bermuda ​for his first visit to ⁠the British territory as sovereign. The Queen will return to Britain.

-with AAP