In the court's biggest civil case to date, the Inuits are demanding the return of their lucrative hunting and fishing grounds plus damages of 234 million kroner (31 million euros, 36 million dollars) for losses incurred since their expulsion in 1953.
The 187 families were forced by the Danish government to leave the village of Dundas, known as "Uummannaq" in Inuit, against their will and with no compensation, and transposed to Qaanaq, more than 100 kilometers (60 miles) to the north, after the United States decided to extend the security perimeter around its Thule radar base.
The base has been one of the key elements of US Arctic defense strategy since the end of World War II, and is now considered an essential part of Washington's plan for a controversial Missile Defense program.
A Danish court ruled in 1999 that Dundas residents would receive 17,000 kroner (2,300 euros, 2,600 dollars) each for their forced move, as well as collective damages of 500,000 kroner (67,300 euros, 78,100 dollars). The Inuits rejected the offer as insufficient.
They then took their case to the supreme court, which began its deliberations on Monday. It is expected to deliver its verdict in mid-November.
A ruling in favour of the Inuits would have international implications, possibly forcing the United States to close the Thule base and putting the government of Denmark -- a strong US ally -- in an awkward position.
However, the Inuits' lawyer Christian Harlang presented a solution that would spare the Danish government such headaches.
"If the supreme court were to rule in favour of the Inuits, and if the Danish parliament were to decide to let the US keep the base, then the hunters should be expropriated" and generously compensated financially, he told Danish daily Berlingske Tidende.
In February 2003, the United States returned a small part of the Inuit hunting grounds in Dundas, an eskimo village founded in 1910 by the Danish explorer Knud Rasmussen.
The deputy mayor of Qaanaaq, Axel Lund Olsen, criticized the terms of that accord reached between Copenhagen and Washington.
"Not only were the citizens of Qaanaaq not heard on the issue, but the Americans left Dundas as it was and have no intention of cleaning up the site after decades of pollution from oil residues, heavy metals and other waste," he said.
In May, Greenland's local government demanded that Washington clean up 54 chemical waste and heavy metal dumpsites polluting the Thule base, even though a 1951 US-Danish defense treaty does not contain any such requirements.
The clean-up call came after the environmental group Greenpeace called the dumpsites a "timebomb" for Greenland and said a 4,000-page classified US report, obtained by Greenpeace, showed that even Washington did not know the full extent of pollution.
Greenland, a Danish colony until 1953, has a population of 56,000, including 48,000 Inuits. After 1953, it became an integral part of Denmark and obtained the status of an autonomous territory in 1979.