Donald Trump snapped back Wednesday at the Danish prime minister's "nasty" dismissal of his attempts to purchase Greenland, heightening a row which has already prompted the US president to scrap a state visit.

Hours after announcing he would not visit Copenhagen next month as planned, Trump accused Prime Minister Mette Frederiksen of insulting the US as a whole by rejecting talk of buying Greenland as "absurd."

With Frederiksen voicing her annoyance at Trump's cancellation, the war of words marks another spat between the US and one of its traditional allies since Trump came to power two years ago on an avowedly "America First" foreign policy platform.

Trump - who made his name as a New York property mogul - has characterized his idea of buying Greenland as essentially "a large real estate" deal, arguing it is a burden on Denmark.

Speaking to reporters at the White House, Trump said he was not the first US president to have raised the idea of buying the vast Arctic island which has housed an American air base since even before it became formally a part of Denmark.

In 1946, President Harry S. Truman offered $100 million in gold, or parts of Alaska, in exchange for Greenland.

"Greenland was just an idea, just a thought. But I think when they say it was 'absurd' - and it was said in a very nasty, very sarcastic way - I said, 'We'll make it some other time,'" Trump said.

"We'll go to Denmark. I love Denmark. I've been to Denmark. And, frankly, we'll do it another time."

The idea of the US buying Greenland was initially dismissed as a joke by some.

The government of Greenland has insisted that the island is "not for sale" and Frederiksen told reporters on Wednesday that she fully endorsed that view.

"I am both annoyed and surprised that the US president has cancelled a state visit," said the prime minister who had been preparing to host Trump early next month.

But, she added, "Denmark and the US are not in crisis, the US is one of our closest allies" and the invitation to visit was still open.