President Donald Trump has attacked Danish Prime Minister Mette Frederiksen, saying she had made "nasty" comments about his interest in having the United States purchase Greenland.

Trump announced Tuesday night that he was abruptly calling off a planned two-day state visit to Copenhagen next month over Frederiksen's refusal to entertain the sale of Greenland, a self-governing country that is part of the kingdom of Denmark.

Frederiksen told reporters Wednesday she was surprised by Trump's change in plans and also lamented the missed opportunity to celebrate the historic alliance between Denmark and the United States, saying preparations for the visit had been "well underway."

Frederiksen called the idea of the sale of Greenland "absurd" over the weekend after news broke of Trump's interest - a characterisation that apparently offended him.

"I thought it was not a nice statement, the way she blew me off," Trump told reporters at the White House on Wednesday. "She shouldn't treat the United States that way... She said 'absurd.' That's not the right word to use."

Trump noted that others had also floated the idea of a US purchase of Greenland, including former president Harry Truman.

Trump's comments Wednesday struck a different tone than Tuesday night, when he said in a tweet that Denmark is "a very special country with incredible people" and he thanked Frederiksen for "being so direct."

....The Prime Minister was able to save a great deal of expense and effort for both the United States and Denmark by being so direct. I thank her for that and look forward to rescheduling sometime in the future!

— Donald J. Trump (@realDonaldTrump) August 20, 2019

Speaking at a news conference in Copenhagen, Frederiksen said Trump's decision to cancel his trip would not "change the character of our good relations," adding that an invitation "for stronger cooperation on Arctic affairs still stands."

Her measured remarks stood in strong contrast with Danish lawmakers from across the political spectrum and former government ministers who slammed the president's behavior as juvenile, undiplomatic and insulting.

"It's an insult from a close friend and ally," Michael Aastrup Jensen, a member of the Danish parliament with the influential center-right Venstre party, told The Washington Post. He said Trump's interest in purchasing Greenland took the country by surprise and was initially widely considered to be a joke, before Danes realised the full extent of "this disaster."

Jensen said Danish lawmakers felt misled and "appalled" by the president, who "lacks even basic diplomatic skills," he said. "There was no word [ahead of time] about: 'I want to buy Greenland and that's why I'm coming.'"

On Twitter, Denmark's former business minister, Rasmus Jarlov, wrote: "For no reason Trump assumes that (an autonomous) part of our country is for sale. Then insultingly cancels visit that everybody was preparing for."

"Please show more respect," he added.

Trump announced the postponement of his visit via Twitter on Tuesday night, writing that Denmark is "a very special country with incredible people," but adding that he had postponed his meeting with Prime Minister Mette Frederiksen after she said "that she would have no interest in discussing the purchase of Greenland."

"The Prime Minister was able to save a great deal of expense and effort for both the United States and Denmark by being so direct," Trump wrote. "I thank her for that and look forward to rescheduling sometime in the future!"

The announcement came two days after Trump told reporters that owning Greenland "would be nice" for the United States strategically. Though the status of the self-governing territory - that is part of the kingdom of Denmark - was initially not publicly cited as a scheduled topic during his visit to Denmark, the postponement of that trip over resistance to his acquisition plans now suggest that it was Trump's central focus in the first place.

Danish officials, including the royal palace, had rushed to organise the presidential visit, which was announced on short notice.

Centre-right lawmaker Jensen called the abrupt cancellation "an insult to the royal house."

Other lawmakers cited by Danish media outlets questioned if the president was still welcome in the country.

Trump's behaviour reminded him of "a spoiled child," Søren Espersen, foreign affairs spokesman for the right-wing populist Danish People's Party, told Danish newspaper Politiken.

"Trump lives on another planet. Self-sufficient and disrespectful," wrote Pernille Skipper, a left-wing Danish politician, on Twitter.

Martin Lidegaard, the chairman of the Danish parliament's foreign policy committee and former foreign minister said in an interview that he hoped Danes would not take this "quite absurd" episode too seriously.

"Understandably, a lot of people are angry," he said, "but we should not let Trump impact Danish-US relations" in a negative way.

"We've been close US allies for decades," he said.

Danes have long considered themselves to have a particularly close relationship with the United States. Denmark actively supported the US-led invasion of Iraq beginning in 2003, even as France and Germany refused to do so. Denmark has also closely collaborated with the United States in the Arctic, which has been a growing focus of the Obama and now of the Trump administration.

After the Cold War, the Arctic lost some of its strategic significance. But climate change and the associated melting of the region's ice cover are making natural resources more exploitable - and turning the Arctic into a region with growing economic and military importance to Russia, China and the United States.

"President Trump's postponement of his visit to Denmark is a setback for our countries' diplomatic relations, but it may be for the best," Anders Fogh Rasmussen, a former Danish prime minister and past head of NATO, wrote on Twitter on Wednesday morning. "The Arctic's security & environmental challenges are too important to be considered alongside hopeless discussions like the sale of Greenland."

Apart from eliciting anger, Trump has had at least one other impact in Denmark in recent days, though. His moves, said center-right lawmaker Jensen, had "strengthened support for Greenland."