Since taking office in January, Trump has vowed to make Greenland, a semi-autonomous territory of Denmark, part of the United States, saying it is vital to U.S. security interests.
Both Russia and China have intensified military activity in the region.
In 2009, it won the right to declare full independence through a referendum, though has not done so on concern living standards would drop without Denmark's economic support.
However, Trump's interest has shaken up the status quo and coupled with the growing pride of the indigenous people in their Inuit culture, put independence front and centre in the election.
In the final debate on Greenland's state broadcaster KNR late on Monday, leaders of the five parties currently in parliament unanimously said they did not trust Trump.
Polling stations open at 1100 GMT and close at 2200 GMT. The final result is expected on Wednesday between 0100 GMT and 0300 GMT. No polls or exit polls are expected.
Reuters spoke to more than a dozen Greenlanders in the capital Nuuk, all of whom said they favoured independence, although many expressed concern that a swift transition could damage the economy and eliminate Nordic welfare services like universal healthcare and free schooling.
However, Greenland has been slow to extract them due to environmental concerns, severe weather, and China's near-total control of the sector, which has made it difficult for companies elsewhere to make a profit or secure buyers.
Trump initially declined to rule out military force, alarming many Greenlanders, although he later softened his stance, stating he would respect the will of the local people and was “ready to invest billions of dollars” if they joined the U.S.
All six main parties, including the ruling Inuit Ataqatigiit and government coalition partner Siumut, support independence but differ on how and when it could be achieved.
“This is our independence election,“ said Qunanuk Olsen, a Naleraq candidate.
According to Egede, the editor, the party could increase its current five seats, but is unlikely to secure a majority in parliament.
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