Prime Minister Malcolm Turnbull has received Donald Trump's personal assurance that a deal for the US to resettle refugees from Nauru and Manus Island will go ahead, despite the US President's harsh immigration policies sending shockwaves around the world.

Fears the deal, struck last year with Barack Obama, might be torpedoed by Mr Trump were allayed during a phone call between the Australian and US leaders on Sunday, Fairfax Media understands.

US President Donald Trump speaks on the phone with Malcolm Turnbull on Sunday morning. Credit:Bloomberg

The assurance came as an executive order signed by Mr Trump clamping down on refugees and broader immigration from some Muslim-majority nations caused large protests in the US. It also prompted concern from Australian MPs from both sides, especially over how the order might affect Australian dual nationals.

Mr Turnbull's office declined to comment on the 25-minute phone call with Mr Trump. Fairfax Media has been told the President confirmed his administration would honour last year's agreement, though it remains unclear how many of the roughly 2000 asylum-seekers held on Nauru and Manus Island will be resettled in the US.