The US president and former Liberal prime minister launched a war of words days before Mr Trump's 25 per cent tariffs on steel and aluminium imports take effect on Wednesday.
While some are concerned the stoush could jeopardise Australia's attempts to negotiate an exemption, former ambassador to the US Arthur Sinodinos has brushed off the criticism.
"Malcolm's a private citizen, which means he's free to speak," he told AAP.
"At the end of the day, it's really the relationship between the prime minister and the president."
Prime Minister Anthony Albanese says he will continue to work in Australia's interests, but the prospect of an exemption appears unlikely.
Australia has argued it should get an exemption because it imports more from the US than it exports - the same successful argument Mr Turnbull used when he was prime minister during Mr Trump's first term.
But members of the president's new administration are fighting harder to ensure no nation is exempt.
"The Americans are very serious about the tariffs - both on revenue grounds and also to encourage more protection of American industry and encourage growth and investment," Mr Sinodinos said.
"The chances are less than 50-50 that we'll get an exemption."
Mr Turnbull conceded the chances of another exemption were low because the administration had regrets over concessions granted in its first term.
He made his third anti-Trump foray in 24 hours on Tuesday morning, urging Australia to stand up for itself.
"We should be free to speak the truth," Mr Turnbull told ABC Radio.
"Or are we going to muzzle ourselves for fear of offending Mr Trump?"
The verbal joust started when Mr Trump posted a spray on social media platform Truth Social on Monday afternoon (AEDT) after Mr Turnbull gave a critical interview to the Bloomberg news service, branding the president rude, erratic and playing into China's hands.
"Malcolm Turnbull, the former Prime Minister of Australia who was always leading that wonderful country from "behind", never understood what was going on in China, nor did he have the capacity to do so," the president wrote.
"I always thought he was a weak and ineffective leader and, obviously, Australian's (sic) agreed with me!!!"
Mr Turnbull lost the prime ministership in a party leadership spill and was not voted out by the public.
The former prime minister previously said the US president was unpredictable and could not be trusted to take military action if Australia, long touted as a trusted and important US ally, was attacked.
Opposition Leader Peter Dutton said Mr Albanese should call the US president immediately to ensure a tariff exemption.
"This is an incredibly important issue for our country, and the priority for me is to make sure that we protect Australian jobs, that we protect Australian industry," he told reporters on the Gold Coast.
Australian Constructors Association chief executive Jon Davies said the looming tariffs represented growing uncertainty for the sector, after being hit with material price rises after COVID and the war in Ukraine.
He said costs associated with steel products could rise because of the tariffs, which would impact construction prices.
"Any additional hit to the construction industry at this time will be extremely problematic," he told AAP.
"It has no contingencies left to weather any significant impacts or downturns."
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