The 31km-wide depression came to light when scientists examined radar images of the island's bedrock.
Investigations suggest the feature was probably dug out by a 1.5km-wide iron asteroid sometime between about 12,000 and three million years ago.
But without drilling through nearly 1km of ice to sample the bed directly, scientists can't be more specific.
"We will endeavour to do this; it would certainly be the best way to get the 'dead fish on the table' (acknowledge the issue, rather than leaving it), so to speak," Prof Kurt Kjær, from the Danish Museum of Natural History, told BBC News.
If confirmed, the crater would be the first of any size that has been observed under one of Earth's continental ice sheets.
The discovery is reported in the journal Science Advances.
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What does the crater look like? The putative impact crater is located right on the northwest margin of the Greenland Ice Sheet, underneath what is known as Hiawatha Glacier.
Additional high-resolution radar imagery gathered by Prof Kjær's team clearly shows a circular structure that is elevated at its rim and at its centre - both classic traits. But because the depression is covered by up to 980m of ice, the scientists have so far had to rely on indirect studies.
What is the supporting evidence? Meltwaters running out from under Hiawatha Glacier into the Nares Strait carry sediments from the depression. In these sediments are quartz grains which have been subjected to enormous shock pressures, of the type that would be experienced in an impact.
Other river sediments have revealed unusual ratios in the concentrations of different metals.
"The profile we saw was an enrichment of rhodium, a depletion of platinum, and an enrichment of palladium," explained team-member Dr Iain McDonald, from Cardiff University, UK.
"We got very excited about this because we realised we weren't looking at a stony meteorite, but an iron meteorite - and not just any old iron meteorite; it had to be quite an unusual composition."
Such metal objects that fall to Earth are thought to be the smashed up innards of bodies that almost became planets at the start of the Solar System.
The signatures identified by Dr McDonald are relatively close to those in iron meteorite fragments collected at Cape York not far from the Hiawatha site. It's not inconceivable, the team argues, that the Cape York material represents pieces that came away from the main asteroid object as it moved towards its collision with Earth.