Prime Minister Jacinda Ardern will tell the Philippine president there have been too many killings in his war against drugs.
Trump's stop in the Philippines will be his last on his 12-day tour of Asia.
After his performance, Duterte said he sang the song for Trump.
"Whoa, whoa", Duterte said, according to a report from the White House press pool.
"We've explained that the United States is open for trade but we want reciprocal, we want fair trade for the United States", he said.
The two countries also vowed to increase cooperation on counterterrorism, intelligence sharing, and will conduct additional exercises.
The relationship between India and the United States goes beyond mutual interest and can be beneficial for Asia and humanity at large, Prime Minister Narendra Modi said on Monday to US President Donald Trump, who described him as a "friend" who is "doing a great job".
More than 3,900 pushers and users have been killed in the war on drugs that Duterte declared when he took office previous year.
Duterte cursed Trump's predecessor, Barack Obama, past year for raising concerns about the war on drugs and he subsequently declared that he was breaking with the United States, a close ally of the Philippines since World War Two.
"President Duterte had several bilateral meetings last night, including with Japan, ROK [Republic of Korea/South Korea], India, and Russia", DFA spokesperson Robespierre Bolivar said in a statement.