Mr McCully is due to meet with President Baron Waqa in Sydney on Thursday on the sidelines of a Pacific Islands Forum meeting of foreign ministers hosted by Australia's Foreign Minister Julia Bishop.
Mr McCully will not rule out cutting New Zealand's aid to Nauru's justice department if its new Government does not take steps to address international concerns about a series of issues in Nauru, including the expulsion of key members of the judiciary.
"New Zealand is taking the reports coming out of Nauru very seriously and this meeting is an opportunity to pass on our concerns and discuss our contribution to the justice sector in light of recent events."
Those events have included the arrests of opposition MPs for attending a protest, seizing passports, banning the use of social networking site Facebook and expelling key figures in the judiciary, since the Government came into power in 2014.
Labour's Foreign Affairs spokesperson David Shearer has called for the Government to take the lead on the issue, saying Australia was compromised by its funding to Nauru for the asylum seekers' centre.
Prime Minister John Key said yesterday that New Zealand would not necessarily only act in coordination with Australia but it would be "useful" if Australia also responded. He said New Zealand's aid was targeted at the justice sector.
"We are not ruling out the fact that we won't change what we are doing. I don't know if we have to be absolutely coordinated in what we do, but a response from both Australia and New Zealand would be useful."
Mr Key said it was unlikely to be part of the UN Security Council work undertaken while New Zealand was chair.
"At this stage, we think we're some way off that. The first instance is to raise our points and put some pressure on them."
The issue is likely to dominate the agenda at the Pacific Islands Forum leaders' meeting in Paua New Guinea in September.
New Zealand's aid to Nauru totals $2.3 million, of which $1.2 million is for its Justice Department. Australia aid is about $AU 21 million ($NZ and will rise to $AU 26 million next year. Australia also funds the asylum seeker processing centre, which has become a major contributor to Nauru's economy.
The Law Societies in both New Zealand and Australia have called for action against Nauru and last week a group of constitutional and international legal academics from New Zealand universities called on the Government to cut New Zealand's $1.2 million aid funding for Nauru's justice department if the rule of law was not restored.