The Washington Post reported Trump abruptly ended the call halfway and described it as the "worst call by far" out of five that day, including Russian President Vladimir Putin.
Trump had reportedly boasted about his election win and objected to Turnbull pushing him to honour an agreement to take 1250 refugees from Nauru and Manus.
Trump, who has suspended refugee processing to the US, was reported as saying he was "going to get killed" politically and that Australia was trying to export "the next Boston bombers".
Turnbull has refused to comment on the reports, but has said Trump agreed to honour the refugee agreement.
The reports paint a very different picture of Trump from his first call to Turnbull, according to Herald editorial writer and Key biographer John Roughan.
Former Prime Minister John Key spoke of a conversation he had with Turnbull to Roughan for an updated edition of the biography John Key - Portrait of a Prime Minister, which is due for release next week.
Key said that shortly after the United States election Turnbull called and told him he had spoken to then President-elect Donald Trump.
Key asked how it went and Turnbull described Trump as "thoroughly professional and businesslike".
When Key then asked what Turnbull thought Trump would be like, Turnbull replied: "Easier to deal with than people think."
The updated version of the biography is due for release next week by publishers Penguin Random House and includes new chapters on Key's final years as Prime Minister and his resignation.
