Gregg and Kathryn Brain, who have lived in Dingwall, in the Highlands, with their son Lachlan, seven, since 2011, were granted leave to remain in the UK until August 1.
Mr Brain and his son came to Scotland as dependants of Mrs Brain, who was on a student visa at a time when a two-year post-study visa was in existence - but it was later abolished.
They hoped a job offer made to Mrs Brain by GlenWyvis distillery in Dingwall would meet visa requirements and allow them to stay.
But the family has since learned the offer has fallen through, giving them just days to find another suitable offer of employment.
Mr Brain said: “In terms of our current situation, we are just praying for a miracle - that someone will come forward with a genuine offer that will satisfy the requirements.
“It really would take a miracle at this point.”
Any job offer would have to meet specific tier two visa requirements, including minimum salary requirements.
New immigration minister Robert Goodwill has said he would be willing to look at extending the family’s leave to remain if a new concrete job offer was made, Mr Brain said.
The Brains’ are being assisted by local politicians, including Ross, Skye and Lochaber SNP MP Ian Blackford.
“We are very grateful for the support and to everyone who has helped us”, Mr Brain added.
GlenWyvis, which launched a crowdfunding drive to finance the distillery, has issued a statement on its Facebook page.
“A genuine offer of administrative role was made to Kathryn Brain in May during our busy crowdfunding,” it said
“No permit to work was reissued sadly and the role was filled, it was also not tier two visa compliant.
“One further job was then also advertised - history curator. Over 40 applications have been received, six are investors/founders.”
In relation to Mrs Brain’s application for the role, it said “information dated July 4 highly relevant to that application, bearing in mind the compressed timeframe, was not sent until July 16”.
“We were advised by MP Ian Blackford it was confidential and he had now been given permission to email it. We have asked Ian Blackford to reason the 12-day delay and are awaiting response.”
The family disputes the statement put forward by GlenWyvis while Mr Blackford said the letter from the then immigration minister James Brokenshire - which may have aided the application process - was sent to the firm on July 6.