The US president-elect can be seen in the footage leafing through a glossy book of photographs while on a jet during his campaign.
Coming to a picture of him and his wife Melania standing alongside the late Queen, he says: “This is with Queen Elizabeth, who was fantastic, by the way.”
The footage has come to light as part of former Fox News host Tucker Carlson’s docu-series Art of The Surge, which reveals some of the behind-the-scenes moments that led to Trump’s decisive victory this week.
He also points to a picture of him and his wife Melania alongside the then-Prince of Wales and Duchess of Cornwall.
“Look, he’s now the King – here’s Charles with the guard. It’s a piece of history at the highest level. Look, it’s Charles. So beautiful.
He adds: “Hopefully he’s going to be well, because he’s a really good person. Camilla is fantastic. You get to know them so well.”
His comments came to light amid concerns about the ability of Sir Keir Starmer’s government to forge a good relationship with the incoming US administration given the previous outspoken criticism of Trump by several key ministers.
The King, who has known Trump for many years, is likely to be asked to play a key role in strengthening ties between the two governments, observers have said.
Trump has previously described the King as “a really wonderful guy”, despite their differences of opinion on climate change.
He visited the UK three times during his first term, which included an inspection of the guard at Windsor Castle with Queen Elizabeth II in 2018 when he famously broke royal etiquette by walking in front of his host.
During a state visit in 2019, which coincided with the 75th anniversary of D-Day commemorations, Trump held talks with the then-Prince of Wales about climate change, which he has previously labelled “a hoax”.
The third visit was in December that year for a Nato summit.
He told the Daily Mail: “We’ve known the King forever. In fact, I was just digging through a photo album not too long ago and I came across a great picture of him, literally with my father. I’m probably 6 years old in the lobby of Mar-a-Lago, in the front entrance of Mar-a-Lago, and it’s special.”