President Trump’s longtime goal of claiming Greenland for America has shifted from rhetoric to official U.S. policy as the White House moves forward on a formal plan to acquire the Arctic island from Denmark.

The plan mobilizes several cabinet departments behind Mr. Trump’s years of talk about wanting Greenland, whose economic and strategic value has grown as warming temperatures melt Arctic ice.

Greenland’s size — 836,330 square miles — also offers Mr. Trump, a former Manhattan developer, the chance to clinch what he may see as one of history’s greatest real estate deals.

Danish officials angrily insist that the sparsely populated island is not for sale and cannot be annexed. But Mr. Trump has made clear his determination to control it.

“We need Greenland for national security and even international security, and we’re working with everybody involved to try and get it,” he said in an address to Congress last month.

“One way or the other, we’re going to get it,” Mr. Trump added.

The White House’s National Security Council has met several times to put Mr. Trump’s words into action, and recently sent specific instructions to multiple arms of the government, according to a U.S. official.

The plan’s full details are unclear. But despite Mr. Trump’s allusions to the possible use of force, the deliberations led by the security council never seriously considered military options, the official said.

The policy instead emphasizes persuasion over coercion, and features a public relations effort aimed at convincing Greenland’s population of 57,000 that they should ask to join the United States.

Mr. Trump’s advisers have discussed using advertising and social media campaigns to sway public opinion on the island, according to another person briefed on the matter.

It may be an uphill battle. In an election last month, an opposition political party that favors quick independence and closer ties with the United States finished in second place but with just a quarter of the vote.

The U.S. messaging campaign will include an unlikely appeal to Greenlanders’ shared heritage with the native Inuit people of Alaska, nearly 2,500 miles away, the official said.

Greenland’s Inuit population is descended from people who migrated from Alaska hundreds of years ago, and the island’s official language is derived from Inuit dialects that originated in Arctic Canada.

Mr. Trump’s advisers have already begun making their public case, arguing that Denmark has been a poor custodian of the island, that only the United States can protect it from encroachment by Russia and China, and that America will help Greenlanders “get rich,” as Mr. Trump has put it.

The Trump administration is also reminding Greenland that the United States has defended it before.

Last month, Mr. Trump posted a slick 90-second video on social media celebrating the “blood and bravery” of U.S. troops who took positions on the island during World War II to prevent a feared Nazi invasion after Germany occupied Denmark.

Although Denmark hoped that American forces would leave after the war, they never did, and the United States still maintains a military base there.

The Trump administration is also studying financial incentives for Greenlanders, including the possibility of replacing the $600 million in subsidies that Denmark gives the island with an annual payment of about $10,000 per Greenlander.

Some Trump officials believe those costs could be offset by new revenue from the extraction of Greenland’s natural resources, which include rare earth minerals, copper, gold, uranium and oil.

Trump officials argue that American capital and industrial might can gain access to the island’s largely untapped mineral wealth in a way that Denmark cannot. “This is about critical minerals,” Mr. Trump’s national security adviser, Michael Waltz, told Fox News in January. “This is about natural resources.”

But analysts do not universally agree that it will be simple to profit from mining in the island’s still-frigid regions. And explaining a significant expenditure to American voters as Mr. Trump has tasked the world’s richest man, Elon Musk, with slashing the federal government by $1 trillion might be tough.

Mr. Trump’s interest in Greenland is not new: He was serious enough in his first term to charge national security officials with exploring the idea. But after he started discussing it publicly, Greenland officials balked and Mr. Trump did not pursue the idea, which was treated as a wild fantasy.

Since his second election last fall, Mr. Trump has renewed his desire with greater fervor. “Let’s get it done,” he has demanded of aides.

“President Trump believes Greenland is a strategically important location, and is confident Greenlanders would be better served protected by the United States from modern threats in the Arctic region,” said the National Security Council spokesman, Brian Hughes.

Mr. Hughes noted that Mr. Waltz and Vice President JD Vance had recently visited Greenland and “laid out the important case for a partnership between Greenland and the United States to establish long-term peace at home and shared prosperity abroad.”

Some analysts say the idea of incorporating Greenland into the United States, or at least developing much closer ties with the island, is less absurd than it might sound.

That is largely because of climate change, which is thawing resource-rich areas and making them more commercially viable. Warmer temperatures have also opened new sea routes through the Arctic for commercial shipping — as well as for Chinese and Russian military vessels.

But Mr. Trump’s vows to control Greenland “one way or another” sound to much of the world like raw imperialism, along with his talk of retaking the Panama Canal and even annexing Canada. If the administration’s persuasion efforts fail, it seems quite possible that Mr. Trump will escalate his tactics.

But that dynamic was noted in December by Robert O’Brien, who served as one of Mr. Trump’s first-term national security advisers.

Denmark, Mr. O’Brien said in an interview with Fox News, “can let us buy Greenland from it, and Greenland can become part of Alaska. I mean, the native people in Greenland are very closely related to the people of Alaska, and we’ll make it a part of Alaska.”

It is unclear how powerfully that message will resonate on the island. While Alaskans share in the profits of their state’s oil wealth in the form of annual checks to residents, its Inuit people endure disproportionate poverty and poor health.

Danish leaders argue that the U.S. pressure campaign is already damaging America’s post-World War II alliance with Denmark.

“We have looked up to you,” Ms. Frederiksen said of the United States during her visit to Greenland this month. “You have inspired us. You have stood guard over the free world.”

“But,” she added, “when you demand to take over a part of the kingdom’s territory — when we are subjected to pressure and threats — what are we to think of the country we have admired for so many years?”