CBS News confirmed Mr. Trump asked his aides and the White House counsel to look into the possibility of buying the island, which is an autonomous Danish territory. It's home to Thule Air Base, the U.S. military's northernmost base.
According to the Wall Street Journal, Mr. Trump proposed the idea multiple times with "varying degrees of seriousness." But Danish politicians have been quick to dismiss it.
"It must be an April Fool's Day joke ... but totally out of season!" former Prime Minister Lars Løkke Rasmussen tweeted.
"No thanks to Trump buying Greenland!" member of Parliament Aaja Chemnitz Larsen said.
"Greenland has been a part of the Danish / Norwegian kingdom for about 500 years before there was anything called the United States at all," member of Parliament Søren Espersen tweeted. "But that kind of detail doesn't interest the trader at the White House. I make myself available for free if he wants a quick history lesson!"
The proposal didn't sit well with former U.S. Ambassador to Denmark Rufus Gifford, who served during the Obama administration and was reportedly considered something of a "rock star" in that country.
"Oh dear lord. As someone who loves Greenland, has been there nine times to every corner and loves the people, this is a complete and total catastrophe," Gifford tweeted.
Mr. Trump is planning his first formal visit to Denmark in early September, for a meeting with the country's new Socialist Prime Minister Mette Frederiksen. Given his purported interest in buying Greenland, politician Uffe Elbæk called the upcoming visit "the most absurd state visit in man's memory."
A White House official with knowledge of the president's trip tells CBS News the White House expects the topic of Greenland to come up during the visit and some staffers are preparing for it.
But the idea of acquiring the island for the U.S. wasn't just dreamed up by Mr. Trump. The U.S. has tried to buy Greenland before. In 1946, former President Harry Truman tried to buy the island from Denmark for $100 million in gold. The offer was rejected and wasn't made known to the world until 1991, after a newspaper in Copenhagen uncovered classified documents detailing the proposal.
Global competition appears to be brewing in Greenland. China has been eyeing the icy territory's minerals and sea routes for years, and now the U.S. and other countries are increasingly taking note of its potential geopolitical power.