Danish Prime Minister Mette Frederiksen has said Greenland is not for sale and Trump's idea of buying it is "an absurd discussion".
Trump's desire to buy Greenland, which is part of the kingdom of Denmark, was first reported last week by the Wall Street Journal.
Two people with direct knowledge of the directive told The Washington Post that the president has mentioned the idea for weeks, and that aides are waiting for more direction before they decide how seriously they should look into it.
White House economic adviser Larry Kudlow confirmed Trump's interest in Greenland, noting that the self-governing country is a "strategic place."
In the days since news broke of Trump's desire to buy Greenland, the idea has been ridiculed by politicians in Denmark, and Greenland's Ministry of Foreign Affairs said Friday that the island is 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," the ministry said in a tweet. "We're open for business, not for sale."
Trump is scheduled to visit Denmark in two weeks, although he said Sunday that his visit is not related to his interest in Greenland.
-AP, Washington Post