Still, the idea of buying Greenland apparently has kept recirculating in his mind. What could be the appeal? Sure, it is the largest island in the world, with (Arctic) ocean views and the world’s only Arctic grass golf course. But there is no private land ownership in Greenland. And it survives in large part through support from Denmark: An annual block grant worth roughly $600 million accounts for more than half of Greenland’s public budget. It doesn’t exactly fit the “Make America Great Again” ethos.

The exact provenance of Mr. Trump’s interest is murky, but one person who has heard the president talk about it said that he has brought up the idea of a possible purchase — albeit fleetingly — since the first year of his presidency. That person, a former senior administration official who spoke on the condition of anonymity to describe private conversations, said Mr. Trump never spoke at length about buying the island, and the context of the conversation centered around the potential of Greenland’s untapped natural resources.

The shrinking ice sheets surrounding Greenland have caught the attention of other Trump administration officials who see potential. Secretary of State Mike Pompeo said in May during an address in Finland to the Arctic Council that he saw new trade opportunities in the Arctic.

“It houses 13 percent of the world’s undiscovered oil, 30 percent of its undiscovered gas, an abundance of uranium, rare earth minerals, gold, diamonds, and millions of square miles of untapped resources, fisheries galore,” Mr. Pompeo said of the Arctic region.

About 80 percent of Greenland is covered in ice, said Jon Gertner, who wrote “The Ice at the End of the World: An Epic Journey into Greenland’s Buried Past and Our Perilous Future.” Although China has expanded its efforts to mine the Arctic, Mr. Gertner said that it remains largely difficult to extract the materials for profit.

“I suppose if you look at the world in terms of natural resources and development, rather than technology and skills,” Mr. Gertner said, “it’s an older, almost antiquated way of looking at things.”