Visitor permits for 35 Indians who claimed to be pilgrims but went missing in Auckland are about to expire.
Forty people disappeared last month while apparently en route to Australia for World Youth Day and the visit of Pope Benedict XVI.
The Department of Labour says four left New Zealand voluntarily and a fifth will leave on Wednesday night.
Of the remaining 35, visitor permits for 16 are due to expire at midnight on Wednesday, and the other 18 at midnight on Thursday.
Once the deadlines pass, anyone remaining without a permit becomes an overstayer and has 42 days to apply to stay in New Zealand.
The department says 10 have already done so by either seeking extensions to their vistor visas or applying for student permits.
It is thought they may have been caught up in a visa scam operating out of the Indian capital Delhi and were led to believe they had permanent residency.