A meeting of officials from the United States, Denmark and Greenland on Wednesday produced no quick fix for the dispute.
"The American ambition to take over Greenland is intact," Danish Prime Minister Mette Frederiksen said in a written comment to Reuters on Thursday, describing a "fundamental disagreement".
"That is of course serious, and therefore we continue our efforts to prevent this scenario becoming a reality."
Trump has said the strategically located and mineral-rich island is vital to US security and that the US must own it to prevent Russia or China occupying it.
He has said all options are on the table for securing the territory, which is an autonomous territory of Denmark.
He says Denmark cannot ward off Russian and Chinese influence in the Arctic region.
Greenland and Denmark say the island is not for sale, that threats of force are reckless and security concerns should be resolved among allies.
Prominent European Union countries have backed Denmark, warning a US military seizure of Greenland could in effect spell the end of the NATO military alliance.
Before Wednesday's meeting in the US, Greenland and Denmark said they had begun to increase their military presence in and around Greenland in co-operation with NATO allies.
Germany, France, Sweden, Norway and the Netherlands have said they are sending military staff to begin preparations for larger drills later this year.
"The Danish Armed Forces, together with a number of Arctic and European allies, will explore in the coming weeks how an increased presence and exercise activity in the Arctic can be implemented in practice," the Danish Ministry of Defence said.
Danish Foreign Minister Lars Lokke Rasmussen said on Wednesday about 200 US troops were stationed in Greenland, which has a population of about 57,000.
The scale of the planned European military build-up has not been made public but initial deployments appear small.
The German Armed Forces were deploying a reconnaissance team of 13, first to Copenhagen, before heading on to Greenland with Danish personnel.
Late on Wednesday, a Danish Air Force plane landed at Nuuk airport and personnel in military fatigues disembarked.
Sweden was sending three officers, and Norway two.
Finland will send two military liaison officers to explore potential training activities, the defence ministry said on Thursday.
Olivier Poivre d'Arvor, France's ambassador to the poles, said France was sending about 15 mountain specialists.
"A first team of French military personnel is already on the ground and will be reinforced in the coming days by land, air and naval assets," French President Emmanuel Macron said.
France and the EU as a whole must be "unyielding in upholding territorial sovereignty," he added.
One United Kingdom officer was joining the reconnaissance group.
The Netherlands said it would send one officer from its navy.
Poland said it would not send soldiers.
After meeting US Secretary of State Marco Rubio and Vice President JD Vance on Wednesday, Rasmussen and Greenland's foreign minister, Vivian Motzfeldt, said the US and Denmark would form a working group to discuss concerns regarding the island.
Greenlandic Prime Minister Jens-Frederik Nielsen reiterated on Facebook on Thursday that the island did not want to be governed by, or owned by, the United States and that it would remain part of Denmark and the NATO alliance.
"Now is not the time for internal discussions. Now is the time for unity, calm and responsibility. I'm following the situation closely, and I stand with you to look after Greenland," Jens-Frederik Nielsen said.
with AP
