Prabagaran Srivijayan was sentenced to death in 2012 for trafficking 22.24 grams (0.8 ounces) of heroin, but has consistently maintained his innocence.
He is expected to be hanged Friday, according to the UN's human rights body which cited family members.
Trafficking certain volumes of illegal drugs carries the mandatory death penalty in Singapore, unless certain conditions are met for it to be commuted to a life sentence.
The UN rights body's Southeast Asia office "calls on the Singaporean government to halt the imminent execution of Malaysian national Prabagaran Srivijayan for a drugs-related offence, and urges the government to immediately instate a moratorium on the use of the death penalty", it said in a statement.
More From This Section
Prabagaran's lawyers have filed a case in Malaysia where the Court of Appeal is considering an application to refer Singapore to the International Court of Justice over concerns about the trial, according to activists.
His legal team has also raised concerns about the fairness of his trial. Amnesty International said this included the alleged failure of the authorities "to follow up leads and call on key witnesses that would corroborate his version of events".
James Gomez, Amnesty International's Director for Southeast Asia and the Pacific, added: "The Singaporean authorities must immediately halt his execution before another person suffers this inhumane and irreversible punishment."
Singapore, however, has consistently maintained that the death penalty is an effective deterrent to crime and has rejected calls to abolish capital punishment.
The city-state last November hanged two foreigners - a Malaysian and a Nigerian - for drug trafficking after their last-minute appeals were rejected.