The Wall Street Journal reported that Trump had expressed interest in the self-governing part of Denmark, which is mostly covered in ice, asking advisors if it would be possible for the US to acquire the territory.
Greenland, located between the North Atlantic and Arctic oceans, is dependent on Danish economic support.
Denmark colonised the 772,000 square-mile (two-million square kilometer) island in the 18th century. It is home to only about 57,000 people, most of whom belong to the indigenous Inuit community.
It also has untapped natural resources like oil, minerals and valuable rare earth elements that China, the United States and other major tech economies covet.
Greenland has been essential to US defense since World War II when it was a base for monitoring Nazi ships and submarines passing through the "Arctic Avenue," the sea gateway to the north Atlantic.
In 1943 the US Air Force built its farthest-north air base at Thule, Greenland.
Thule was crucial in the Cold War, the first line of monitoring against a potential Russian attack.
With a population of 600, the base today is part of the NATO mission, operating satellite monitoring and strategic missile detection systems and handling thousands of flights a year.
"Luckily the US is able to ensure and meet its security interests by maintaining this air base in northern Greenland. There's no requirement to buy Greenland to keep America safe."
"The early warning radar system in northern Greenland helps protect North America and is a key part of our missile defense apparatus," said Luke Coffey of The Heritage Foundation.
Yet as the polar ice sheet began to shrink, the Russians became more active and China has moved to establish itself in the region.
US Secretary of State Mike Pompeo underscored the revived US interest in a speech in May in Finland, where he slammed China and Russia for "aggressive behavior" in the Arctic.
"The region has become an arena of global power and competition" owing to vast reserves of oil, gas, minerals and fish stocks, he warned.
"Just because the Arctic is a place of wilderness does not mean it should become a place of lawlessness," he said.
With no geographical claim to the region, but whose massive commercial shipping industry would benefit from new polar routes as the ice melts, China is the newcomer whose presence could shift the balance.
It began sending scientific missions in 2004. In the past several years, a Chinese company has gained mining rights for rare earths, partnering with an Australian company in the Kvanefjeld project.
In January 2018 Beijing unveiled its "Polar Silk Road" strategy to extend its economic footprint through the Arctic.
To gain favour in Nuuk, the Chinese have wined and dined government officials.
China's role in the Arctic has been more about expanding its economic influence, soft power.
"It's hurting Denmark very badly because they're losing almost $700 million a year carrying it... and strategically, for the United States, it would be nice," Donald Trump said.
When asked if he would consider trading a US territory for Greenland, Trump replied that "a lot of things could be done."
"Essentially, it's a large real estate deal," he added.
Greenland's ministry of foreign affairs insisted Friday the island was ready to talk business, but was not for sale.
"#Greenland is rich in valuable resources such as minerals, the purest water and ice, fish stocks, seafood, renewable energy and is a new frontier for adventure tourism," it tweeted.
"We're open for business, not for sale," it added.