Greenland voted on Tuesday in legislative elections that could yield a timeline for independence for the Danish self-governing territory coveted by US President Donald Trump.

Possibly signalling a Trump effect, turnout was higher than usual in the sole polling station in the capital Nuuk, according to election officials who extended voting - which had been due to end at 8:00 pm - by half an hour to allow long lines of voters to cast their ballots.

First results were expected several hours later.

Trump, who has said he is determined to get his hands on the vast Arctic island "one way or the other", tried until the last minute to influence the vote, sparking astonishment, rejection, and, to a small degree, enthusiasm among the 57,000 Greenlanders, most of whom favour independence.

After waiting in a long line with other voters, Greenland's outgoing Prime Minister Mute Egede - who celebrated his 38th birthday on Tuesday - cast his ballot shortly after polling statinos opened.

"Our country is in the eye of the storm," he said in a video posted to Facebook just hours before the vote.

"The international community is watching us closely, and we have recently seen how much they are trying to influence our country," said Egede, leader of the left-green party Inuit Ataqatigiit (IA).

The lead-up of the election to choose the 31-seat parliament, the Inatsisartut, was mostly marked by debates on healthcare, education, and future ties with Denmark, which still controls foreign, defence and monetary policy.

Greenland's inhabitants - almost 90 percent of whom are Inuits - say they are tired of being treated like second-class citizens by their former colonial power, which they accuse of having historically suppressed their culture, carried out forced sterilisations and removed children from their families.

All of Greenland's main political parties back independence but disagree on the timeframe. (AFP)