The president said he wants to be “very involved” with the UK because “you don’t hear the word Britain any more”.
He made the comments after tweeting that a US-UK trade deal could be “very big & exciting”.
In an interview with the Wall Street Journal, Mr Trump declined to give any detail on how a post-Brexit transatlantic agreement agreement may look.
He added: “But I can say that we’re going to be very involved with the UK. I mean, you don’t hear the word Britain any more. It’s very interesting. It’s like, nope.”
The president, who has Scottish ancestry, also expressed concern over a second Scottish independence referendum.
He said: “It would be terrible. They just went through hell.”
The owner of luxury golf courses in Ayrshire and Aberdeenshire, Mr Trump appeared to be worried about the future of the British Open in the event that the Scottish nationalists won.
“What would they do with the British Open if they ever got out? They’d no longer have the British Open,” he told the newspaper.
Mr Trump spoke to the newspaper on July 25, when he tweeted: “Working on major Trade Deal with the United Kingdom. Could be very big & exciting. JOBS! The E.U. is very protectionist with the U.S. STOP!”
His comments came as International Trade Secretary Liam Fox was in Washington for talks with US trade representative Robert Lighthizer.
The meeting came after critics raised fears that British markets could be opened up to US agricultural products currently blocked by EU food standards rules, including controversial chlorine-washed chicken.
The UK Government has said Brexit offers an “unprecedented opportunity” to reshape Britain’s trading ambitions, although what that may mean for aspects such as food and agriculture has already proved controversial.