Trump, who takes office on January 20, has described US control of Greenland, a semi-autonomous Danish territory, as an "absolute necessity".
He did not dismiss the potential use of military or economic means, including tariffs against Denmark.
Axios reported on Saturday, citing two sources, that the Danish government wanted to convince Trump that his security concerns could be addressed without claiming Greenland.
A spokesperson for the Trump transition team did not respond to a request for comment on the Axios report.
Trump's eldest son, Donald Trump Jr visited Greenland last week.
Danish Prime Minister Mette Frederiksen said that she had asked for a meeting with Trump, but did not expect it to happen before his inauguration.
Greenland Prime Minister Mute Egede too said he was ready to speak with Trump but urged respect for the island's independence aspirations.
Denmark has previously said that Greenland is not for sale.
Greenland has a population of 57,000 and has vast natural resources including oil, gas and rare earth minerals.
Australian mining company Energy Transition Minerals filed a lawsuit in 2023 after Greenland stopped its project to extract rare minerals on the Arctic island.
