The Prime Minister says Mr Trump has "absolutely not" asked for anything in return for US help in resettling up to 1250 refugees languishing on Manus Island and Nauru, despite making it clear during the pair's now infamous phone call – and publicly via social media – that he dislikes it.

Mr Turnbull's latest comments came as Mr Trump vowed to fight and win an appeal against a judge's order that lifted his controversial ban on immigrants from seven Muslim-majority nations.

But in a blow to Mr Trump's hopes of a quick resolution, a federal appeals court late on Saturday denied the Justice Department's request for an immediate reinstatement of the ban.

Despite the continued anti-refugee rhetoric out of Washington, Mr Turnbull believes Mr Trump's assurance that he will proceed with the deal Australia struck with the Obama administration still stands.

Mr Trump reportedly branded it the "worst deal in the world" during the leaders' heated exchange last weekend – details of which were later leaked to the media – then saying on Twitter it was "dumb".

He also accused Mr Turnbull of seeking to export "the next Boston bombers" into the US.

"It's a deal obviously that President Trump has said he wouldn't have entered into but he has committed to honour it," Mr Turnbull told veteran journalist Laurie Oakes during an interview on the Nine Network's 60 Minutes on Sunday.

Oakes then pressed Mr Turnbull on whether that would make Australia indebted to the Trump administration if it sought to launch another Middle Eastern military adventure or moved to act against China in the South China Sea.

"We assess all requests for military assistance on their merits and there is no linkage at all between an arrangement relating to refugee resettlement and any other matters," Mr Turnbull said.

He still refused to talk about the details of last Sunday's phone call, saying only it was "frank and forthright". He said the controversy over the call – which had Mr Trump accused of bullying US allies – had only highlighted the strength of the US-Australia relationship.

"I don't think there has ever been more public support for Australia than there has been this week," Mr Turnbull said. "We have seen dozens and dozens of congressmen and senators talking about the importance of the Australian alliance and talking about what a great friend we have always been to the United States.

"So this has been a very good week for Australia."

Mr Turnbull also laughed off a series of gaffes by White House spokesman Sean Spicer, who repeatedly referred to Mr Turnbull as "Mr Trunbull" or "Mr Trumble".

Mr Trump on Saturday launched a scathing personal attack on US District Judge James Robart, whose ruling lifted his temporary ban on refugee admissions.

"The opinion of this so-called judge, which essentially takes law-enforcement away from our country, is ridiculous and will be overturned!" Mr Trump said on Twitter.

"The judge opens up our country to potential terrorists and others that do not have our best interests at heart. Bad people are very happy!"

Judge Robart, appointed by former Republican president George W. Bush, questioned the constitutionality of Mr Trump's order.

Judge Robart said no attacks had been carried out on US soil by individuals from the seven countries affected by the travel ban. For Mr Trump's order to be constitutional, he said it had to be "based in fact, as opposed to fiction".

The court ruling was the first move in what could be months of legal challenges to Mr Trump's anti-immigration push, which set off chaos last week at airports across the US where travellers were stranded and thousands gathered to protest.

Late on Saturday in the US, a federal appeals court denied the Justice Department's request for an immediate reinstatement of the travel ban.

The 9th US Circuit Court of Appeals in San Francisco instead asked challengers of the ban to respond to the appeal filed by the Trump administration, and for the Justice Department to file a counter-response by Monday afternoon.

The court's denial of an immediate stay means the legal battles will continue for days at least.

The sudden reversal of the ban catapulted would-be immigrants back to airports, with uncertainty over how long the window to enter the US would remain open.

The Department of Homeland Security said it would return to its normal procedures for screening travellers but that the Justice Department would file for an emergency stay of the order "at the earliest possible time".

The travel ban, which Mr Trump said was needed to protect the US against Islamist militants, sparked travel chaos around the world and condemnation by rights groups who said it was racist and discriminatory.

With Reuters, AP, The Washington Post