One senior Republican colleague, senator John McCain, even took the step of calling the Australian ambassador to reaffirm the alliance on Thursday.

Speaking at the annual National Prayer Breakfast in Washington, Mr Trump strayed from his prepared remarks several times and addressed the intense media coverage of his phone call with Mr Turnbull - which turned sour during a discussion over an Obama-era agreement to take 1250 refugees from Australia's offshore detention camps - as well as an equally controversial conversation with the Mexican President Enrique Peña Nieto.

"When you hear about the tough phone calls I'm having, don't worry about it," Mr Trump said. "Just don't worry about it. They're tough. We have to be tough."

"It's time we're going to be a little tough folks. We're taken advantage of by every nation in the world virtually. It's not going to happen anymore. It's not going to happen anymore."

Revelations about the call with Mr Turnbull were followed by a tweet from the president condemning the "dumb deal" on refugees and pledging that he would "study" it on Wednesday night. The tweet caused further confusion coming shortly after the US Embassy in Australia issued a statement saying Mr Trump's decision to "honour the refugee agreement has not changed".

That confused message continued on Thursday, though the White House seemed to suggest the refugee deal would be honoured.

Asked whether the deal was indeed going ahead on Thursday, White House press secretary Sean Spicer suggested the "extreme vetting" of refugees was indeed going ahead, while also appearing to suggest it was still under review.

"The president is unbelievably disappointed in the previous administration's deal that was made and how poorly it was crafted and the threat to national security that it put the United States on," Mr Spicer, who described the conversation as "cordial", said on Thursday afternoon.

"He has tremendous respect for the prime minister and the Australian people, and has agreed to continue to review that deal and to ensure that as part of the deal, which was always part of it, we would go through a very, very extreme vetting process to ensure that every single person that is being offered is coming here with peaceful intentions and poses no threat to the United States."

Mr Spicer appeared to mispronounce Mr Turnbull's name during the press conference, calling him "Mr Trunbull" or "Mr Trumbull".

Shortly after, Mr Trump defended the phone call again in off-the-cuff remarks during an event in Washington, but said if the previous administration did something "you have to respect that" - seeming to concede he would honour the deal.

He said the deal involved "probably well over a thousand illegal immigrants who were in prisons and they were going to take them into this country and I said why? Just wanted to ask the question."

"1250 [people], could be be 2000, could be more than that. I said why, why are we doing this? What's the purpose? So we'll see what happens."

"Previous administration does something you have to respect that but you can also why are we doing this?"

The controversy prompted mixed responses from the president's Republican colleagues. Senator McCain, a former Republican presidential candidate who has clashed with Mr Trump on issues of foreign policy before, described the president's treatment of Australia as "harmful."

"It was an unnecessary and frankly harmful open dispute over an issue which is not nearly as important as United States-Australian cooperation and working together, including training of our marines in Australia and other areas of military cooperation and intelligence," he said during a doorstop interview in Washington DC.

Mr McCain later put out a statement saying he had personally called Australia's ambassador to the US, Joe Hockey, to express support for the alliance.

"I asked Ambassador Hockey to convey to the people of Australia that their American brothers and sisters value our historic alliance, honour the sacrifice of the Australians who have served and are serving by our side, and remain committed to the safer, freer, and better world that Australia does far more than its fair share to protect and promote," he said in a statement.

House Speaker Paul Ryan however downplayed the incident on Thursday, saying that Australia was a central ally, "they are and they will continue to be."

"I don't think Australia should be worried about its relationship with our new president or with our country for that matter," he said.

Republican senator Lindsey Graham, who has also been a frequent critic of Mr Trump, played down the impact of the call on the alliance though, saying "the relationship is strong" during an interview on CNN.

On the president's tweet about Australia he conceded "I wish he would sleep more and tweet less... there's probably better ways to handle this."

Republican senator Bob Corker also tweeted on Thursday afternoon that he had telephoned Mr Hockey to discuss the "important and long-lasting alliance between our two countries."

Kellyanne Conway, a counsellor to the president, denied the leak of the phone calls had come from within the White House.

"It's very unfortunate," she said of the report during an interview on Fox News. "Obviously we're not commenting... we're the ones not leaking..."

But she later added "anyone who is just discovered that President Donald J. Trump is a resolute, decisive man who doesn't mince his words and who is putting America and her allies first and her people and her interest first, is waking up I think out of the cave."

Mr Trump also used his address at the prayer breakfast to attack his successor as the host of The Apprentice, actor and former Republican governor of California Arnold Schwarzenegger, urging the gathered religious leaders to pray for Schwarzenegger's ratings.

Schwarzenegger fired back on Twitter soon after, offering to swap jobs with the president - allowing Mr Trump to go back to television and "allowing people to sleep comfortably again."