Senator Graham, Chairman of the Senate Judiciary Committee and an outspoken ally of Mr Trump, said he will write to the Australian, UK and Italian governments urging them to co-operate with US Attorney General Bob Barr's investigation.

Mr Barr is looking into what prompted the FBI and other intelligence agencies to investigate the Trump campaign before the 2016 election win.

"This New York Times article about Barr talking to Australia is the beginning of an effort to shut down Barr's investigation to find out how this whole thing started," Graham told FOX News on Monday.

The New York Times put Australia in the spotlight on Monday when it reported Mr Trump "pushed" Mr Morrison in a recent phone call to help investigate the origins of US Special Counsel Bob Mueller's inquiry.

The newspaper also reported the White House restricted access to the call's transcript to a small group of presidential aides.

Mr Trump has been under fire the past week and faces an impeachment inquiry for pressuring another key ally, Ukraine, to investigate Democrat presidential rival Joe Biden.

The Australian government swiftly moved to quell any controversy by confirming Mr Morrison was willing to help with Mr Barr's probe.

"The Australian government has always been ready to assist and co-operate with efforts that help shed further light on the matters under investigation," the spokesman told AAP on Tuesday.

"The PM confirmed this readiness once again in conversation with the president."

The Australian government's eagerness to show public support for Mr Trump comes after Mr Morrison's recent visit to the White House for a state dinner and a side trip to Ohio with the president where the two leaders attended a campaign-like rally at Australian billionaire Anthony Pratt's new paper recycling plant.

Mr Trump jokingly praised Mr Morrison as a "man of titanium".

Australia has maintained a close relationship with the temperamental president despite the nation playing a key role in sparking the 2016 FBI probe into links between the Trump campaign and Russia.

Mr Mueller confirmed in his report its was a London bar meeting between Australia's then UK high commissioner Alexander Downer and Trump foreign relations adviser George Papadopoulos that prompted the FBI to open the probe.

Mr Downer told The Australian newspaper last year Mr Papapadopoulos said during the drinks session Russia had damaging material on Mr Trump's presidential rival Hillary Clinton.

Mr Downer forwarded the information back to Australia and it was passed on to US authorities.

Mr Papadopoulos, who claims he was being spied on by Australian, UK and US intelligence agencies, told AAP he did not tell Mr Downer about the highly-sensitive information during their bar meeting.

Mr Trump said on May 24 earlier this year he hoped Mr Barr would look "at the UK and I hope he looks at Australia and I hope he looks at Ukraine".

The comment led Australian Ambassador to the US Joe Hockey to write a letter four days later to Mr Barr and White House Acting Chief of Staff Michael Mulvaney.

"The Australian government will use its best endeavours to support your efforts in this matter," Mr Hockey, who has golfed with Mr Trump, wrote in the May 28 letter.

"While Australia's former High Commissioner to the United Kingdom, the Hon. Alexander Downer, is no longer employed by the government, we stand ready to provide you with all relevant information to support your inquiries."

Australian Associated Press