The US president 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.
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Allegations of Russian meddling in last year's US presidential elections 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, including at another regional summit in Vietnam and a banquet dinner in Manila on Sunday night, appeared to support Duterte's confidence that Trump was not concerned with the killings.
"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.
'I already killed someone'
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 personally killing people, 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.
Duterte also said in December last year that he had personally shot dead criminal suspects when he was mayor of southern Davao city to set an example for the police.
Then-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".
Trump has so far 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 their brief chat in Vietnam the previous day.
jpegMpeg4-1280x720And at the pre-summit banquet on Sunday, Duterte sang a Filipino love song in front of his audience, saying in a light-hearted fashion that he did so on the orders of the US president.
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.
Competing territorial claims in the South China Sea and fears that the Islamic State group is gaining a foothold in Southeast Asia are expected to be on the formal agendas of the talks.
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