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Rahaf Mohammed Alqunun had planned to enter Australia on a tourist visa and seek asylum before she was detained by Thai authorities on Sunday.
The 18-year-old is currently in the care of UN officials who say it will take about five days to process her request for assistance, after she said she feared her family would kill her if she were sent home.
Health Minister Greg Hunt says the Australian government has been urging the UN High Commissioner for Refugees to process her case quickly.
"We've been successful in getting them to agree to do that," Mr Hunt told ABC TV on Wednesday.
Australia will consider giving Ms Alqunun a humanitarian visa if the process finds she is a refugee, he says.
"Pending the outcome of that, if she is found to be a refugee, then we will give very, very, very serious consideration to a humanitarian visa."
Ms Alqunun's father and brother arrived in Bangkok on Tuesday evening and have asked to see her.
But Thai immigration chief Surachate Hakpan said the men would have to wait to learn whether the UN's refugee agency would allow the request.
"The father and brother want to go and talk to Rahaf but the UN will need to approve such talk," General Surachate told reporters.