The Danish government has said Mark Rutte cannot negotiate on behalf of Denmark or Greenland over the future of the Arctic island, as the broad outlines of a deal apparently struck by the Nato secretary general and Donald Trump began to emerge.

The US president said after meeting Rutte on Wednesday the “framework of a future deal” had been found to settle the transatlantic dispute over Greenland following weeks of escalating tensions that risked the biggest breakdown in relations in decades.

Speaking on Thursday, Denmark’s prime minister, Mette Frederiksen, welcomed the progress that had been made but said the situation was still “difficult and serious”.

Pointedly, she added: “It is only Denmark and Greenland themselves that can make decisions” on issues concerning them. “We can negotiate all political aspects – security, investment, the economy – but we cannot negotiate our sovereignty.”