Greenland tells Trump it is not for sale
Trump's controversial remarks came hours after he announced that he intended to nominate Ken Howery, his former ambassador to Sweden, to be the new ambassador to Denmark.
Mr Howery said he was "deeply humbled" by the nomination and looked forward to working with the staff at the US embassy in Copenhagen and the US consulate in Greenland to "deepen the bonds between our countries".
Trump's original suggestion in 2019 that the US acquire Greenland, which is the world's largest island, led to a similarly sharp rebuke from leaders there.
The then Danish Prime Minister Mette Frederickson, who still holds the role, described the idea as "absurd", leading Trump to cancel a state trip to the country.
He is not the first US president to suggest buying Greenland. The idea was first mooted during the 1860s under the presidency of Andrew Johnson.
Separately on Sunday, Donald Trump threatened to reassert control over the Panama Canal, one of the world's most important waterways - accusing Panama of charging excessive fees for access to it.
Panama's president later said "every square metre" of the canal and surrounding area belonged to his country.