Mr Goff said the detention issue had made headlines, but the current rules affecting hundreds of thousands of law-abiding Kiwis in Australia were "patently unjust".
"One case referred to me, a man had a brain tumour, then was not entitled to a sickness benefit. He got health treatment, because that is reciprocal, but...he actually wasn't in a position to travel back to New Zealand because of his sickness."
Prime Minister John Key has this week revealed his view that Australia might soften its position and give the estimated 250,000 to 350,000 Kiwis living there on "non protected" visas a realistic pathway to citizenship.
Both Labour and National-led Governments have been lobbying for change since Kiwis' rights were greatly reduced in February 2001 by John Howard's Liberal Government.
The new rules meant Kiwis arriving to live in Australia were issued with special visas, under which they have no welfare safety nets in case of injury or misfortune, cannot vote and have no automatic path to permanent residency or citizenship.
Mr Goff told the Herald that when the Australian Government was looking to introduce the new rules, it had asked New Zealand to pay $1 billion a year to cover expat Kiwis' welfare costs. Australia was also worried New Zealand was being used as a back door by migrants from the Pacific and Asia.
New Zealand Treasury provided information that showed Australia took in much more than that in tax from the group.
"We said no," Mr Goff said. "That would have been Australia double-dipping...in the end, the only agreement we could reach, apart from superannuation, was that it was up to each country to take the action that they thought necessary."
Mr Little and Mr Goff will present at two select committees in Canberra on the issue, as well as meet with Immigration Minister Peter Dutton and Opposition leader Bill Shorten.
Mr Goff said he was encouraged that the Australian Labor Party had at its annual conference earlier this year adopted a resolution that points to giving expat Kiwis a path to citizenship.
"I think what has changed is I don't believe there is any great sentiment of bludging Kiwis there at the moment."
There are signs that Malcolm Turnbull's Government may be considering a shift in position.
Mr Key said this week that Mr Turnbull had raised the issue with him, and was also aware of New Zealanders' unhappiness at the deportation of criminals, many of whom had lived in Australia most of their lives.
"If he can't immediately fix that [the deportee] issue, he may be able to fix the way that would stop them actually being deported in the first place," Mr Key said, adding that many Kiwis in Australia were in a "state-less" position.
In an October visit to New Zealand, Mr Turnbull announced new legislation that would give New Zealanders who came to Australia as dependent children and had lived there for at least 10 years access to student loans.
