COPENHAGEN — President Trump’s idea to possibly buy Greenland for its natural resources left residents of the semiautonomous Danish territory amused, apoplectic and in disbelief, and received a chilly reception in Denmark on Friday.

“I hope it’s a joke, because it’s a terrible and grotesque thought,” said Martin Lidegaard, chairman of Denmark’s Foreign Policy Committee.

“It must be an April Fool’s Day joke … but totally out of season,” Lars Lokke Rasmussen, a former prime minister of Denmark and the leader of the opposition, posted on Twitter.

The idea first sprang up last year, when Mr. Trump was said to have joked about buying Greenland for its natural wealth during a meeting in the Oval Office. He is said to have repeatedly returned to the possibility, since the country, which is part of the kingdom of Denmark, appeals to him because its location in the North Atlantic has security value, according to people familiar with his thinking.

His advisers were highly skeptical that a purchase of the world’s largest island could ever happen, but they agreed to investigate the possibility.

“Greenland is not for sale and cannot be sold, but Greenland is open for trade and cooperation with other countries — including the United States,” Kim Kielsen, Greenland’s premier, said in a statement, according to the Ritzau news agency.

The foreign minister, Ane Lone Bagger, and the Twitter account of the Ministry of Foreign Affairs echoed that sentiment.

The report is likely to add an unexpected element to Mr. Trump’s planned state visit in less than three weeks to Denmark to meet with Prime Minister Mette Frederiksen, Queen Margrethe II and the leaders of Greenland and the Faroe Islands. Greenland and the Arctic are high on the agenda.

The Danish prime minister’s office did not immediately return a call on Friday seeking comment.

Social media users were quick to exploit the report about the American president’s exploring the purchase of Greenland. One photoshopped a pompous-looking golden tower into a picture of Greenlandic villages with colorful two-story wooden houses. Another asked if Denmark could trade Greenland for Hawaii.

But at the center of the international contretemps, there was little to laugh about, politicians and residents said.

“It’s never nice to be treated as a commodity,” said Aaja Chemnitz Larsen, a Greenlandic member of the Danish Parliament.

Greenland, a nation of 56,000, has a shared history with Denmark since the first Vikings settled there a millennium ago. If that relationship were to change, it would not be up to Denmark and certainly not up to an American president’s “impulse,” said Henrik O. Breitenbauch, an expert on Greenland and the head of the Center for Military Studies at Copenhagen University.

“You don’t just trade people and countries,” Mr. Breitenbauch added.

The speculation is that Mr. Trump, a former real estate developer, was keen on Greenland because he tends to see the world through a prism of acquisitions. And Greenland, located among both friendly and hostile neighbors, has everything a real estate investor could desire in terms of fresh air, direct access to the sea, an abundance of shrimp, cod and halibut and a backyard rich with lucrative minerals.

Nearby international sea routes allow for quick passage to all corners of the globe (when the ice permits). But the island’s population may see little to gain from exchanging the Danish queen as their head of state with an American president who has angered traditional allies by disparaging NATO and pulling the United States out of long-held treaties like the Paris climate accord.

“Greenland could choose to become Puerto Rico with snow, but I doubt there’s much interest in that,” Mr. Breitenbauch said, referring to the unincorporated United States territory.

The country also has more pressing worries: Climate change has accelerated the melting of Greenland’s mile-thick ice sheet, leaving behind lots of sand. If the entire ice sheet melted, it would raise sea levels by about 20 feet, studies show.

In recent years, the Danish government has asserted its influence over Greenland to block Chinese investments out of concern for potential Greenlandic dependence on China. The Danish involvement has caused friction with Greenlandic leaders, who have denounced it as neocolonialism.

Although Greenland now has its own government with vast autonomy, its foreign and defense policies originate in Copenhagen, the Danish capital, as does part of its national budget: Greenland receives $740 million annually from Denmark.

Greenland has vital strategic importance to Denmark, but it’s also an integral part of Danish history and its self-image as a nation of explorers and sailors.

“It’s been a space for exploration, discovery and part of the narrative of the creation of the modern Danish nation,” Mr. Breitenbauch said. “We would become a much more parochial part of Europe without.”

Though Mr. Trump’s buyer’s interest may have little foundation in reality, it does point to a sensitive spot in Greenland’s quest for independence: Where but from Danish taxpayers would it get 50 percent of its budget?

“There’s a discussion in Greenland about the cost of independence and how to pay for that,” said Rasmus Kjaergaard Rasmussen, an expert on Greenland and Arctic affairs at Roskilde University in Denmark. “Either they sell a lot fish, find oil in the Arctic region, find uranium or other minerals and tax them. Or they find a new Denmark, a new sponsor to cover the $740 million. That could be the United States.”

He added that while American sponsorship was an unlikely outcome, to even discuss it could help Greenland residents clarify for themselves where they could find a substitute for the millions sent from Denmark every year.

Martin Breum, the author of “The Greenland Dilemma,” said Mr. Trump’s idea is “an absolutely radical break” with settled foundations since World War II. “It’s questioning a power relation more than 70 years old,” he added. “When small nations wake up to the world’s superpower threatening to unroot that relation, it’s not something to take lightly.”

“This may be a signal that the United States wants a different relationship with Greenland than what we’ve been used to,” he mused. “If that’s the case, it’s fundamental to the way the Danish realm works. Greenland comprises 99 percent of the realm’s territory. It’s everything but a joke,” Mr. Breum said.

“Even if this is an aphorism, it’s paramount to clarify whether this has more to it. If there is a desire to change the existing. It has enormous importance to Greenland,” he said.

Aki-Matilda Hoegh-Dam, a member of the Danish Parliament elected in Greenland, wondered whether Mr. Trump had been speaking for his own amusement. “It sounded like something we shouldn’t take too seriously although it’s extremely provocative,” she said.

But she added: “It’s important for us to point out that selling Greenland is not an option. Nobody can sell a country like that. Denmark doesn’t own Greenland and you can’t sell something you don’t own.”

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