Ioane Teitiota and Angua Erika and their three New Zealand-born children have lost claims for refugee status due to global warming.
Teitiota has been battling deportation to Kiribati since 2011, when he overstayed his visa.
He argued his family faced indirect persecution from man-made climate change, which has caused rising sea levels, contaminated water, destruction of crops, tidal surges, and extreme weather.
But at his last appeal in July, the Supreme Court ruled while Kiribati undoubtedly faced challenges, Teitiota did not face serious harm if he returned there.
The family's lawyer, Michael Kidd, is asking for a stay of any deportation order until they find out the result of an appeal to the United Nations' Human Rights committee.
“Teitiota's moral argument is that he is blameless in the destruction of his country, and there needs to be a recognition by surrounding countries such as NZ that these people need to be offered some sort of asylum or refugee status,” Kidd said.
He said it was the last legal avenue for the family, who feared their children would be susceptible to illness because they had never lived in Kiribati to develop immunities.
Kidd said the family was nervous and insecure, and hoped the Government would take a compassionate approach to their case.