An online petition to keep them in the country claims the family have spent tens of thousands on a degree to keep them in Scotland.
"Now, without right to appeal, their passports have been confiscated, their right to work abolished, and the Home Office have threatened to freeze their bank accounts and cancel their driving licenses," it states.
"They do not deserve to be treated like criminals."
"All they want is for the Home Office to deliver what was promised when they sold everything they owned in Australia to come to the UK - a two year post-study work visa, giving them the opportunity to establish a working relationship with an employer, and a chance at a longer-term visa. Living up to what was promised is not too much to ask."
The government later said the family face no "imminent risk of immediate deportation" from Scotland to Australia.
Immigration Minister James Brokenshire offered assurances in parliament over the future of the family, who live in Dingwall in the Scottish Highlands.
Mr Brokenshire confirmed he is planning to discuss the case further with SNP MP Ian Blackford.
He said he had twice extended the time available for the Brain family to secure the necessary permission for them to remain in the country.
"We've not yet received an application from the Brain family for leave to remain under the points-based system but we will consider any application they make," he said.
One of the key issues in the debate on Britain's membership of the European Union ahead of a June 23 vote is the arrival of immigrants seeking work, and their status as beneficiaries of Britain's welfare system.
Immigration is above targets set by the government, seen in data released on Thursday showing net migration to Britain rose to 333,000 last year. The number of new arrivals from Europe has driven much opposition to the bloc.
"The government apparently is trying to regulate immigration but what they are actually doing is alienating and deporting the very people with the talent and the skills that we need in Scotland," SNP MP Kate Forbes said.
"These inflexible rules are a blunt instrument," she said.
Gregg Brain is a health and safety expert and Kathryn, who arrived on a student visa, has just completed a degree in Scottish history and has an offer of temporary work in a local distillery.