In November, Canberra negotiated a "one-off" deal with the outgoing Obama administration to settle an unspecified number of the 1,600 boatpeople Australia held in offshore processing centres in Nauru and Papua New Guinea.
There were fears that Trump, who on Friday signed an executive order to suspend the arrival of refugees to the US for at least 120 days and bar entry for 90 days to people from seven Muslim-majority countries, would scupper the deal.
Trump suspends refugee arrivals to ‘make US safe’
Following a phone call between the leaders Sunday, Australia Prime Minister Malcolm Turnbull said the agreement would proceed.
"We also discussed the resettlement arrangement of refugees from Nauru and Manus, which had been entered into with the previous administration, and I thank President Trump for his commitment to honour that existing agreement," Turnbull told reporters.
He declined to provide further details on the deal, saying it was for US authorities vetting people on the islands.
"So this is a matter that is entirely in the hands of United States government's agencies."
Turnbull did not join in with international criticism of the Trump travel ban, saying he would not comment on the policies of sovereign nations.
Amid confusion in the US, where immigration authorities have struggled to implement the ban, thousands have protested at several airports.
Around 300 people were stopped or detained worldwide, while four federal judges moved to halt deportations.
Trump says new order on refugees is not a Muslim ban
The US order suspends the arrival of Syrian refugees indefinitely and bars citizens from Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen for 90 days.
Canberra has come under fire from rights groups and the United Nations for its treatment of people seeking to arrive by boat, many from war-torn areas.
The government puts such boatpeople in offshore detention camps and blocks their resettlement in Australia, even if found to be refugees.
Immigration Minister Peter Dutton said such policies would continue.
"I want to be very clear, though, today to people smugglers and people who would seek to exploit this announcement into an opportunity to put more people onto boats, that the government's resolve remains as strong as it's ever been," he told reporters of the US deal.
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