Trump offered to buy the vast Danish territory during his first term in office -- receiving an abrupt refusal -- and he revived his push over the weekend when naming his ambassador to Copenhagen for his incoming administration.
Greenland's Prime Minister Mute Egede quickly sought to quash any chance of a deal.
"Greenland is ours. We are not for sale and will never be for sale. We must not lose our long struggle for freedom," Mute Egede said in a statement.
Greenland, the world's largest island, is an autonomous Danish territory with its own parliament, about 55,000 inhabitants, and a small pro-independence movement.
It relies on Denmark to fund more than half of its public budget.
Trump on Sunday posted that "for purposes of National Security and Freedom throughout the World, the United States of America feels that the ownership and control of Greenland is an absolute necessity."
As president, he canceled a state trip to Denmark in 2019 after the country said Greenland was not for sale.
Under President Joe Biden, the United States said it did not want a purchase but instead sought to strengthen ties.
