Labour immigration spokesman Rajen Prasad said the ongoing threat to deport Mr Kumar was "cruel and heartless" because kidney transplants could not be done in Fiji.
"What I want the government to do is show more compassion and allow this young man to receive the transplant that will not cost the tax payer anything," Dr Prasad said.
Immigration Minister Michael Woodhouse said the decision on Jooste was made by the Immigration and Protection Tribunal and not Immigration NZ, which was fighting the delayed deportation of the South African.
Being unlawfully in New Zealand was the reason for deporting 1621 migrants, followed by 69 whose circumstances no longer met visa rules and 43 for criminal offending.
Natalie Gardiner, INZ acting compliance operations manager, said an interview to establish a migrant's individual circumstances before a deportation decision is made.
Those liable for deportation may initially be held in custody for up to 96 hours, but this was considered on a case-by-case basis.
"It is important to stress that the majority of those unlawfully in New Zealand are not criminals, they choose to remain here for a number of reasons such as employment and family," she said.
"In these cases, INZ focuses more attention on ... managing them towards what is termed as a voluntary departure."
Those of leave voluntarily would have to pay their own costs but did not face a ban from returning to New Zealand.
The number of voluntary departures had increased over recent year rising from 1166 in 2010/11 to 1392 in 2011/12 and 1773 last year.
Mr Woodhouse said the removal costs have halved in the past seven years, while the number of overstayers had also dropped significantly.
"The figures show a really good outcome in recent years, voluntary departures are increasing markedly, total removal costs are down, total overstayer numbers are lowest in recent history," he said.
Immigration estimates showed there were 13,151 overstayers as at last Sept, compared with 15,880 in 2009 and 20,328 in 2004.
Paola Benedicto, who last month spoke to the Herald when her husband Alexis Benedicto was deported back to Chile, said she was puzzled at how deportations were decided.
She and her 6-year-old daughter were granted interim visas and were not deported along with her husband, who did not have a visa.
"Why do they let me stay here when they kick my husband out and break up our family?" said Mrs Benedicto, who has since left New Zealand.
Migrant Action Trust manager Agnes Granada said many deportation decisions "seemed unfair" and immigrants from some countries appeared to be favoured more than others.
Deportation (over the last 5 years)
• $7,167,815 - deportation cost to Nov 2013
• 4,525 - deportations
• 13,151 - overstayers in NZ today
Facing deportation:
Sanil Kumar, from Fiji: needs urgent kidney transplant, but given one week to make arrangements to depart.
Hendrik Pieter Jooste, from South Africa: former Auckland City Council employee who stole $350,000 of ratepayer money but won a four-year stay after a tribunal appeal - which is being challenged by INZ.
