US President Donald Trump ripped into his Australian counterpart in a phone call last week, reports said, castigating an Obama-era refugee deal he later described on Twitter as "dumb".

The Washington Post said Trump abruptly cut short the fiery conversation after criticising the agreement to resettle people kept in Pacific camps, sparking a war of words with Australian Prime Minister Malcolm Turnbull on Thursday after the report surfaced.

Australia is considered a close US ally - one of the so-called "Five Eyes" with which the US routinely shares sensitive intelligence - and the call might have been expected to be smooth sailing.

But, according to the Post, Trump's assessment was the opposite.

Of his four conversations with world leaders that day, "this was the worst call by far," it cited him as telling Turnbull, shortly before he terminated the telephone meeting.

Australian government sources told the Australian Broadcasting Corporation the report was "substantially accurate".

Turnbull said he was disappointed details of the "very frank and forthright" exchange had been leaked.

"As far as the call is concerned I'm very disappointed that there has been a leak of purported details of the call in Washington," he told Sydney radio station 2GB.

"But I want to make one observation about it - the report that the president hung up is not correct. The call ended courteously."

READ MORE: Nauru - A place of abuse and desperation

He added that Canberra had "very, very strong standards in the way we deal with other leaders and we are not about to reveal details of conversations other than in a manner that is agreed".

The Post's account is markedly different from the official version of the call provided by both governments.

Turnbull said on Monday that Trump had agreed to honour the deal agreed with then president Barack Obama to resettle an unspecified number of the 1,600 people Australia holds in offshore detention centres on the islands of Nauru and Papua New Guinea.

There were fears the new US president would rescind it after he signed an executive order last week to suspend the arrival of refugees to the US for a least 120 days, and bar entry for three months to people from seven Muslim-majority countries.

Diplomatic crisis

After the Post story broke late on Wednesday, Trump weighed in on Twitter and threw the agreement into doubt.

Do you believe it? The Obama Administration agreed to take thousands of illegal immigrants from Australia. Why? I will study this dumb deal! — Donald J. Trump (@realDonaldTrump) February 2, 2017

After seeing the tweet, Turnbull insisted that Trump had already agreed it would go ahead.

"The commitment made by the president in that call was made and we announced that and it was confirmed by his spokesman a day or so later," he said.

Reports of the Turnbull-Trump conversation came as the US and Mexican governments were denying claims that the former reality TV star had threatened to send troops into Mexico to deal with drug cartels.

Mexican journalist Dolia Estevez, citing "confidential" US and Mexican sources, said Trump made the threat during an hour-long phone call with President Enrique Pena Nieto on Friday.

Mexican and White House officials vehemently denied the report.

Relations have plunged into the biggest diplomatic crisis between the two neighbours in decades.

Trump angered Mexicans last week by ordering the construction of a massive border wall and vowing to make their country pay for it.

Pena Nieto has pledged that his government will never pay for the barrier and cancelled a meeting with Trump scheduled for this week in Washington.

In addition to the row over the wall, Trump wants to renegotiate the North American Free Trade Agreement with Canada and Mexico. Mexican officials expect talks to begin in May.