John McCain called Turnbull later to reiterate the United States' commitment to the alliance with Australia - but the White House staff-produced transcripts the newspaper obtained show how the conversation unfolded.
The conversation between the two leaders grew sour as Trump rejected an agreement to take refugees.
Mr Trump did have praise for Australia's tough treatment of refugees arriving by boat, calling it a "good idea" and telling Mr Turnbull "you are worse than I am" - meant as a compliment.
Trump told Turnbull on January 28 that he had been making calls to other world leaders and that one was his worst. Turnbull insists to Trump that the deal is still on. And since January 27, a federal investigation has been launched into potential Russian interference in the 2016 presidential race and how that might have benefitted Trump.
Trump called the law - identified as his among top campaign priorities - 'the least important thing we are talking about, but politically this might be the most important'.
"I have to have Mexico pay for the wall - I have to".
Trump goes on to tell Nieto that can't tell the media that Mexico won't pay for the wall.
Donald Trump and Mexican President Enrique Peña Nieto at the G-20 Summit in Hamburg, Germany, on July 7.
"Because", Trump continued in his telephone call, "you and I are both at a point now where we are both saying we are not to pay for the wall".
During the call, the president placed much of the blame for America's drug crisis on Mexican drug lords who "are sending drugs to Chicago, Los Angeles, and to NY".
The state's two Democratic senators, Maggie Hassan and Jeanne Shaheen, on Thursday criticized Trump's description of their state and the way in which he appeared to refer callously to the opioid crisis.
Republican Sen. Lindsey Graham, an occasional critic of the president's, said it was unfair that the leaked transcripts were being shared. "It's disappointing his mischaracterization of this epidemic ignores the great things this state has to offer", Sununu said in a statement.
Trump's call with Turnbull, which only lasted 24 minutes, was more hostile.
"Boy, that will make us look awfully bad", Trump says.
'Let me tell you, Mr. President, this is not a personal difference. "That would make us very even with Mexico and it would make a lot of sense".
Trump agreed, adding that when asked by the press about the border wall, he would respond, "Let us see how it is going - let us see how it is working out with Mexico".
"I hate taking these people".
"There is nothing more important in business or politics than a deal is a deal", he says.