Denmark’s prime minister has told Donald Trump to “stop the threats” about taking over Greenland.

Mette Frederiksen said “it makes absolutely no sense to talk about the need for the United States to take over Greenland”, adding: “The US has no right to annex any of the three nations in the Danish kingdom”.

Her remarks come after Katie Miller – the wife of one of Trump’s aides, Stephen Miller – tweeted a map of Greenland in the colours of the American flag alongside the word “SOON”.

Trump has repeatedly raised the possibility of Greenland becoming an annexed part of the US, citing its strategic location and mineral wealth.

In her statement, posted on the Danish government website, Frederiksen said she was addressing the US “very directly”.

She said Denmark – “and thus Greenland” – was a Nato member and covered by the alliance’s security guarantee. Denmark already had a defence agreement with the US that gave it access to Greenland, she said, and Denmark had upped its investment in security in the Arctic region.

“I would therefore strongly urge the United States to stop the threats against a historically close ally and against another country and another people who have very clearly said that they are not for sale,” she said.

Earlier the Danish ambassador to the US responded to the post by Ms Miller – a right-wing podcaster and former aide to Trump during his first term – with a “friendly reminder” that the two countries were allies and saying Denmark expected respect for its territorial integrity.

The exchange comes after the US carried out a major military operation against Venezuela on Saturday, capturing its president Nicolas Maduro and his wife and removing them to New York.

Trump later said the US would “run” Venezuela and US oil companies would “start making money for the country”.

The US president has previously refused to rule out using force to secure control of Greenland, a vast island in the Arctic.

Trump has claimed that making it part of the United States would serve US security interests due to its strategic location and its abundance of minerals critical to high-tech sectors.

The Trump administration’s recent move to appoint a special envoy to Greenland prompted anger in Denmark.

Greenland, which has a population of 57,000 people, has had extensive self-government since 1979, though defence and foreign policy remain in Danish hands.

While most Greenlanders favour eventual independence from Denmark, opinion polls show overwhelming opposition to becoming part of the US.