The royal household had formally invited the US president to visit Denmark on 2-3 September.
The president reportedly became interested in the island, which is a semi-autonomous territory of Denmark, after hearing about Greenland’s natural resources.
But the 73-year-old insisted at the weekend that such a transaction was not a top priority.
Denmark had rebuffed the idea, saying Greenland was not for sale.
The US has not purchased land in such a way since 1917, when the country bought the US Virgin Islands.
The president’s idea was also widely mocked online.
“Greenland is not for sale, Greenland is not Danish, Greenland belongs to Greenland,” Mette Frederiksen said over the weekend.
“I strongly hope that this is not meant seriously,” she said to the Sermitsiaq newspaper.
She also dismissed the discussion as “absurd.”
“Denmark is a very special country with incredible people, but based on prime minister Mette Frederiksen’s comments, that she would have no interest in discussing the purchase of Greenland, I will be postponing our meeting scheduled in two weeks for another time,” Mr Trump tweeted on Tuesday.
The royal palace’s spokesperson said the household had no further comments.
There was no immediate reaction from the Danish government.
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Additional reporting by agencies