"Four people are missing," local broadcaster KNR quoted local police chief Bjorn Tegner Bay as telling a news conference in the autonomous Danish territory.
There were no confirmed fatalities, but Bay said 11 houses had been swept away after a magnitude 4 overnight quake off Uummannaq, a small island well above the Arctic Circle.
"The huge waves risk breaking over Upernavik and its environs. The residents of Nuugaatsiaq are going to be evacuated," police said on Facebook, referring to nearby hamlets.
"A good explanation is that the quake created a fault at the origin of a tsunami," meteorologist Trine Dahl Jensen told Danish news agency Ritzau, warning of potential aftershocks.
"It's not normal, such a large quake in Greenland," she said.
KNR quoted Ole Dorph, mayor of Qaasuisup, a municipality in the area affected, as lamenting "a serious and tragic natural catastrophe which has affected the whole region."
Danish Prime Minister Lars Lokke Rasmussen tweeted news of what he termed a "terrible natural catastrophe at Nuugaatsiaq."
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