Kathryn and Gregg Brain, who moved from Australia to Dingwall in the Scottish Highlands with their son Lachlan in 2011, had until the end of Monday to secure a job for Mrs Brain that meets Home Office visa requirements, after being granted an extension on their leave to remain.
Their local SNP MP Ian Blackford said he has received a letter from the Home Office saying the Brains would not get indefinite leave to remain or a further extension to the grace period, after the last extension lapsed on August 1.
It means they are now illegal immigrants and if they do not leave of their own accord they face arrest.
The family initially moved to the UK on Mrs Brain's student visa but a two-year post-study visa scheme then on offer was later withdrawn by the Government.
Mr Brain said the family have received around a dozen job offers but none that meet the specifications for a UK Tier Two visa.
Mr Blackford MP posted on Facebook: "My duty over the coming days and weeks is to do all I can to assist the family, not coerce the family to leave.
"I will continue to work over the coming days with the family to find a suitable opportunity that will satisfy the tier two requirements. The challenge we all face is still to unearth an opportunity that will allow the family to stay."
Mr Brain said their local church had offered the family support.
In the letter to the MP, also sent to the Brain family and to Scottish External Affairs Secretary Fiona Hyslop, UK Immigration Minister Robert Goodwill said: "We need to treat the Brain family fairly, but also everyone else in a similar position.
"There is no fundamental difference between their circumstances and that of any other individuals who came to the UK on a temporary study visa and there are no exceptional considerations which would justify granting them leave outside the immigration rules.
"Their case will now be handled by the family returns unit of immigration enforcement.
"Dedicated Scotland-based family engagement staff will contact the family later this week to begin discussions with them regarding a voluntary departure to Australia. These conversations will focus on how much time and what help and assistance the family need to make this a reality.
"I very much hope you will encourage the family to co-operate with this process and to leave the UK voluntarily."
A Home Office spokesman said: "We have not received any fresh application from or on behalf of the Brain family which would allow them to stay in the UK.
"We have given the family three extensions on an exceptional basis over a number of months to allow them to try to secure a job that would allow them to meet the immigration rules, but this cannot be open-ended.
"In line with established policy designed to apply evenly and fairly to everyone, anyone who is unable to regularise their stay is expected to leave the UK voluntarily."
