The New York Times, citing two unidentified US officials, earlier reported the request to assist in the investigation -- which aims to discredit the Russia probe -- was made during a recent telephone call between Trump and Morrison.
An Australian government spokesman confirmed on Tuesday that the request was made, saying in a statement Australia "has always been ready to assist and cooperate with efforts that help shed further light on the matters under investigation".
"The PM confirmed this readiness once again in conversation with the President."
The Times said the White House had restricted access to the transcript of the call, in a similar way to its handling of Trump's recent call with Ukrainian President Volodymyr Zelensky.
That call -- during which Trump asked Zelensky to investigate his potential 2020 Democratic presidential opponent Joe Biden -- has led to the opening of an impeachment inquiry in the Democratic-led House of Representatives.
The Washington Post reported separately that Attorney General Bill Barr has had multiple meetings overseas with foreign intelligence officials in an effort aimed at undermining US intelligence's conclusion that Russian interference helped Trump to victory in the 2016 presidential election.
In addition to contacting the Australian leader, Barr has met with British intelligence and last week was in Italy, together with US Attorney John Durham, asking for Rome's help in revisiting the 2016 Russia investigation, according to the newspaper.
The Post said Barr risked appearing to use his powers as head of the US Justice Department to help Trump politically when he is facing impeachment and, potentially, removal from office for abuse of power.
DOJ probing 'political witch hunt'
Former special counsel Robert Mueller's probe into Russian interference in the US election was completed in March with the identification of numerous acts of collusion or attempted collusion between the Trump campaign and Moscow.
But after Mueller ruled there was not enough material for criminal charges, Trump called for another investigation into what he called the Russia "political witch hunt" and "hoax."
The Justice Department chose Durham to examine the roots of the Mueller probe, including the evidence that led the CIA, FBI and other intelligence agencies to declare that Russia did in fact interfere in 2016 to boost Trump.
Australia and Britain were both important because the original FBI Russia meddling probe was launched in July 2016 after the bureau received a tip from an Australian diplomat in Britain about a meeting between people tied to Russian intelligence and a Trump campaign official.
But Barr's involvement in the Durham investigation has raised questions that it is a partisan political probe.
In a statement, the Justice Department defended the contacts abroad and said Trump's and Barr's role was to help Durham make contacts.
"Mr. Durham is gathering information from numerous sources, including a number of foreign countries," said Justice Department spokesperson Kerri Kupec.
"At Attorney General Barr's request, the president has contacted other countries to ask them to introduce the attorney general and Mr. Durham to appropriate officials," she said.