Gregg Brain, 49, and his 48-year-old wife Kathryn, have been at the centre of a high profile immigration row.
The couple, who live at Dingwall in Ross-shire, have a seven-year-old son, Lachlan, who has been educated in a local primary school and speaks Gaelic.
Scotland's First Minister Nicola Sturgeon has called plans to deport the family "outrageous" and has complained to the UK Government, which controls immigration.
The Brains moved to the United Kingdom in 2011 from Brisbane in Australia, entering on a student visa granted to Kathryn to study Scottish history and archaeology.
They had hoped to stay in the country by taking advantage of the post-study visa scheme, which allowed non-EU graduates from a British university or college to remain in the UK for at least two years.
However, the British Government cancelled the scheme in 2012, closing off that particular visa "pathway".
Now the family has to apply for a Tier 2 visa, which requires at least one of the parents to have a certain category of job, which neither has.
A recent, well-publicised job offer to Kathryn from a local distillery fell through.
Without the offer of a job and the associated sponsorship of an employer, the Brain family faces deportation.
They have been granted an extension to their grace period in their attempts to find work but that runs out at midnight on Monday.
Mr Brain told Sky News: "The situation, as it stands now, is that we are basically praying for a miracle.
"The injustice that we've suffered is that a visa pathway that was open to us and promised to us was torn up after we committed to coming here.
"We've invested six figure sums in the local economy, we've paid over ?40,000 in taxes and contributed to the local community."
There will be no instant deportation when the clock strikes midnight on Monday.
However, barring any last-minute reprieve, the Brain family will be expected to make arrangements to leave the country voluntarily.
A Home Office spokesman said: "All visa applications are considered on their individual merits, and applicants must provide evidence to show they meet the requirements of the immigration rules."
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