John Key will hold a meeting with him in Auckland tomorrow, and will urge him to lower the number of New Zealand citizens being detained and deported from Australia.
However, any concessions will be a hard-sell with many Australian voters and members of Mr Turnbull's own Liberal Party.
Mr Key's language this week has reflected that - he has talked up the shared Anzac history, called Mr Turnbull "sensitive" and denied the possible deportation of 1000 Kiwis is a strain on the relationship.
Opposition parties have been blunter. The Greens' spokeswoman for human rights, Catherine Delahunty, yesterday said Mr Turnbull should not jeopardise the Anzac spirit by sticking with his government's "obsession with political point-scoring on immigration".
And Labour leader Andrew Little, who will also meet Mr Turnbull today, said there had been two camps of Australian prime ministers in recent history. Those such as Paul Keating and John Howard clearly valued the relationship with New Zealand while Bob Hawke, Kevin Rudd and Tony Abbott did not.
"The question for Malcolm Turnbull is, is he a New Zealander-phile Australian prime minister, or is he not," Mr Little said "We want to see what he is. As I understand it, it wouldn't require a change to the law... but it is fair to ask that a more sensible level of discretion is exercised when the policy is being applied."
A law change late last year - backed by the Australian Labor Party - has meant non-Australians who incur a prison sentence of a year or more face deportation, and has led to about 240 Kiwis being held in seven Australian detention centres, including on Christmas Island.
Mr Turnbull said yesterday the issue was "a very live one", but it was important to "maintain our standards, our security" and discretion was used in applying the law.
Previewing Mr Turnbull's visit, ABC News political editor Chris Uhlmann observed the new Prime Minister "looks set to be caught between the rock of Mr Key and the very hard place of his colleagues".
"Being seen to back down on any aspect of border protection will only confirm the worst fears of the conservatives," Mr Uhlmann wrote.
One of Mr Key's pitches will be to point out that New Zealanders in Australia are "lifters, not leaners".
"As a group we are more employed than Australians as a group, we earn more than average Australians as a group, we are actually incarcerated at a lower rate than Australians as a group," the PM said this week.
If no ground is given, more funding will have to be made available to help integrate deportees, many of whom have no connection with the country of their birth.
