Former Prime Minister Tony Abbott and his wife, Marg, visited Auckland the year before, although he was rolled later that year.
Key previously hosted former Labor Prime Minister Julia Gillard and her partner, Tim Mathieson, to a bilateral visit to Queenstown in 2013, although she was rolled later that year.
Disenchantment with Turnbull has begun to emerge from his own party, which prompted a more aggressive style of attack from him last week against Labor leader Bill Shorten.
The future of the Trans Pacific Partnership in the wake of the United States' withdrawal is expected to be high on the agenda - and the pair may compare notes about their phone encounters with US President Donald Trump.
Trump was said to be very grumpy at the prospect of having to honour a deal done by Turnbull and Barack Obama for the US to take 1250 asylum seekers currently in detention in Papua New Guinea and Nauru.
English said he and Turnbull would discuss global economic trends "and our shared interest in advancing the trade agenda in the Asia Pacific."
"We will continue discussion on the situation of New Zealanders in Australia, particularly following Prime Minister Turnbull's welcome announcement in 2016 of a pathway to citizenship."
In a rule change announced in Sydney a year ago, thousands of Kiwis who arrived in Australia after it tightened its immigration rules in 2001 will be given an easier path to seek Australian citizenship under certain conditions.
If they earned A$53,000 over five consecutive years ($57,000) between 2001 and a year ago, they will eventually be able to apply for permanent residence and eventually apply for citizenship.
Key estimated that up to 100,000 of the 305,000 Kiwis who arrived to live in Australia between 2001 and a year ago could meet the criteria and get a new pathway to citizenship.
Treatment of New Zealanders who arrived in Australia after 2001 had been a growing point of contention in an otherwise close Transtasman relationship.
The changes in 2001 withdrew several types of benefit (unemployment, youth and sickness) for New Zealanders - who enter Australia on a Special Category Visa.
But the most significant change was to make it much harder to get citizenship.
It required Kiwis to compete with other immigrants for permanent residence - which is capped - instead of being able to skip that and apply directly for citizenship as they had previously been able to.
