Greenland is strategically located between Europe and North America, making it a critical site for the US ballistic missile defence system.
Its rich mineral resources also fit Trump's goal of reducing US dependence on China.
The island is an autonomous territory of the Kingdom of Denmark.
It has its own parliament and government but Denmark retains authority over foreign affairs and defence.
"We encourage our current (Greenlandic) government actually to have a dialogue with the US government without Denmark," said Pele Broberg, the leader of Naleraq, the largest opposition party and the most prominent political voice for Greenland's independence.
"Because Denmark is antagonising both Greenland and the US with their mediation."
Naleraq, which strongly advocates a rapid move to full independence, doubled its seats to eight in last year's election, winning 25 per cent of the vote in the country of just 57,000 people.
Although excluded from the governing coalition, the party has said it wants a defence agreement with the US and could pursue a "free association" arrangement - under which Greenland would receive US support and protection in exchange for military rights, without becoming a US territory.
All Greenlandic parties want independence but differ on how and when to achieve it.
Greenlandic Foreign Minister Vivian Motzfeldt said Greenland could not conduct direct talks with the US without Denmark because it is not legally allowed to do so.
"We must respect the law, and we have rules for how to resolve issues in the Kingdom," she told Sermitsiaq daily.
The Danish and Greenlandic governments did not immediately reply to requests for comment on Broberg's remarks.
The comments come ahead of a planned meeting between the Danish and Greenlandic foreign ministers and US Secretary of State Marco Rubio next week to address tensions between the NATO military alliance allies.
Motzfeldt said it was important to set Greenland's relationship with the United States on a steady course.
"My greatest hope is that the meeting will lead to a normalisation of our relationship," she told Sermitsiaq.
Denmark and Greenland are seeking to steer the debate back onto a diplomatic track, where rational concerns can be addressed, said Ulrik Pram Gad, senior researcher at the Danish Institute for International Studies.
"The strategy is to get Trump's undiplomatic social media statements put back into stabilised diplomatic channels," he told Reuters.
In Brussels, NATO ambassadors held a "cordial" discussion on Greenland on Thursday, with broad agreement that the alliance should beef up its Arctic security efforts, according to people familiar with the talks.
"No drama," a senior NATO diplomat said.
"Lots of agreement that NATO needs to accelerate its development of stronger deterrence presence in the region."
Exactly what additional steps NATO might take remains undecided.
Some countries have suggested using the alliance's operations on its eastern flank - Baltic Sentry and Eastern Sentry - as potential models.
Both missions deploy international forces flexibly to protect several countries while employing drones, sensors and other technology to monitor land and sea.
