Prime Minister Jacinda Ardern is being urged to make her offer to take refugees directly to Nauru rather than Australia.
World Vision New Zealand has launched a campaign to resettle 119 children and their families who are detained on Nauru.
Ardern has previously offered to take 150 refugees from Manus and Nauru - a move that was rejected by Australia, who claimed it would open back-door entry into that country.
Ardern left the offer on the table.
World Vision national director Grant Bayldon said they were asking Ardern to make the offer directly to Nauru, rather than Australia.
"While there remains an obligation on Australia under international law, it doesn't look like they are going to do the right thing any time soon."
Another element of the campaign was that they were asking for the evacuation of children and their families to be prioritised, Bayldon said.
They should be brought to safety in New Zealand before Universal Children's Day in November, he said.
"These vulnerable people seeking refuge and asylum should be resettled in New Zealand as part of an emergency intake over and above our refugee quota," he said.
Bayldon said the children and their families need to be moved urgently so they can access support and rebuild their lives in dignity.
"They are being forced to live in sub-standard conditions which violate the United Nations Convention on the Rights of the Child.
"They lack adequate healthcare, education, and protection from sexual and physical abuse."
New Zealand had the capacity and heart to take these kids off Nauru, he said.
The other thing that had happened since the Prime Minister's offer was that President Donald Trump had put a stop to citizens from seven countries travelling to the United States, he said.
"My understanding is that impacts about half the refugees on Nauru."
In a statement, Ardern said no one liked the thought of children being forced to flee their homes and seek asylum, "no matter where they end up and where in the world that happens".
"New Zealand's offer to Australia to take 150 refugees from Manus Island and Nauru has been repeated a number of times and remains on the table.
"New Zealand has also previously discussed refugee-related issues with Nauru. The Government of Nauru has made clear publicly that its focus is on working with Australia on appropriate resettlement options."
World Vision New Zealand is asking Kiwis to take the plea to the Prime Minister on Facebook using the hashtag #KidsOffNauru ahead of the Government visit to the country.
The New Zealand Government will attend the Pacific Island Forum in Nauru on September 3.