President Donald Trump has repeatedly said the United States needs to annex the Danish autonomous territory to ensure national security.
"A foreign individual is offering money for a signature to incorporate Greenland into another country. That is not just deeply worrying. It is indecent," Nielsen wrote on Facebook.
Greenlandic broadcaster KNR on Thursday reported that a "mysterious American" calling himself "Cliff" had offered a taxi driver in Greenland's capital Nuuk $200,000 to sign the petition.
The taxi driver, Danny Brandt, told KNR he declined the offer.
Brandt wrote a social media post about his encounter, and a person in the comments section said he had also been offered money to sign.
Brandt said he reported the incident to police.
Greenland police told AFP they had "received reports that cannot be ruled out as being connected to the current political situation."
They would not confirm how many reports they had received nor their "substance".
Prime Minister Nielsen said: "We are a democratic society. Our future is not negotiated in a taxi. And it is not bought with money."
"Decisions about our country are made by us. That ought to be easy enough to understand," he said.
Trump in January backed down from several intense weeks of aggressive rhetoric, announcing that he had reached a "framework" agreement on Greenland with NATO's secretary general, the details of which remain vague.
Denmark and Greenland are in the process of holding talks with Washington about the future of the Arctic island.
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