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.