Before taking office for his second White House term, Mr Trump said he would not rule out using military force to seize control of the island, calling it vital to US national security.
Promising to send "Lego executives and the cast of Borgen" to make the deal, the petition to turn Los Angeles into "Løs Ångeles" adds that "rule of law, universal health care, and fact-based politics might apply".
A light-hearted petition has been launched in Denmark to buy the state of California in after Donald Trump said he wanted to take over Greenland.
During his first term as US president, Mr Trump discussed the possibility of claiming Greenland, which is a semi-autonomous Danish territory, and a vast and mineral-rich Arctic island.
At a joint news conference in Copenhagen last month, Ms Frederiksen and Mute B Egede, Greenland's Prime Minister, rejected the plan.
