The US leader is in Manila with leaders of 18 other nations for two days of summits, the final leg of a headline-grabbing Asian tour dominated by the North Korean nuclear crisis.
Allegations of Russian meddling in last year's US presidential elections have also hounded the second half of his 12-day trip, which took him from Japan to South Korea, China and Vietnam.
Rights groups have called on Trump to end his Asian journey with a strong statement against Duterte's drugs war, which has seen police and suspected vigilantes kill thousands of people.
But brief encounters between them in the lead-up to official talks scheduled for late Monday morning appeared to support Duterte's confidence that Trump was not concerned with the killings.
Trump shook hands with Duterte, then the pair chatted for about 30 seconds as the Philippine leader smiled broadly, before the opening ceremony for the first summit on Monday morning. Trump had his back to the cameras.
The pair also sat next to each other at a pre-summit banquet on Sunday, during which they smiled, chatted and clinked champagne glasses.
Duterte, 72, sang a Filipino love song in front of his audience at the banquet, saying in a light-hearted fashion that he did so on the orders of the US president.
"I'm sure he will not take it up," Duterte said on Sunday when asked whether he expected Trump to raise the issue of alleged extra-judicial killings in the drugs war.
Duterte won elections last year after promising to eradicate illegal drugs with an unprecedented campaign that would see up to 100,000 people killed.
Since he took office, police have reported killing 3,967 people in the crackdown.
Another 2,290 people have been murdered in drug-related crimes, while thousands of other deaths remain unsolved, according to government data.
Many Filipinos back Duterte, believing he is taking necessary measures to fight crime, but rights groups warn he may be orchestrating a crime against humanity.
Amnesty International accuses police of shooting dead defenceless people and paying assassins to murder addicts.
When pressured over allegations of extra-judicial killings carried out by police, Duterte insists he has never told them to break the law.
But rights groups say police are following Duterte's incitements to kill, including comments made last year when he said he would be "happy to slaughter" three million addicts.
He has also repeatedly boasted about killing people himself, most recently on Thursday while in Vietnam for the Asia-Pacific economic summit.
"At the age of 16, I already killed someone. A real person, a rumble, a stabbing. I was just 16 years old. It was just over a look," Duterte said.
Former US president Barack Obama was one of many prominent critics of Duterte's handling of the drugs war. The Philippine leader responded last year by calling Obama a "son of a whore".
But Trump has appeared to be a fan of Duterte, telling him in a telephone call in April that he was doing a "great job".
Duterte said on Sunday that Trump had offered him further "words of encouragement" during a brief chat in Vietnam the previous day on the sidelines of another regional summit.
Duterte is hosting the world leaders because the Philippines holds the rotating chair of the 10-nation Association of Southeast Asian Nations (ASEAN) bloc.
The events on Monday and Tuesday in Manila are two separate ASEAN-hosted summits, which also include China, Japan, Russia, South Korea, India, Canada, Australia and New Zealand.
The rising threat of the Islamic State group across Southeast Asia, and further efforts to pressure North Korean leader Kim Jong-Un to abandon his nuclear ambitions, were top agenda items in Manila.
"Terrorism and violent extremism endanger the peace, stability and security of our region because these threats know no boundaries," Duterte said in an opening ceremony speech on Monday.
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)