The Brain family, who have been living in Scotland since 2011, must leave the UK and return to Australia in six days' time if they cannot meet visa criteria.
Gregg and Kathryn Brain's son Lachlan is a pupil in Gaelic medium education.
The UK government said the family, from Dingwall, had twice been given extra time to help them meet visa rules.
Moray SNP MP Angus Robertson has also challenged Chancellor George Osborne on the Brains' situation.
During Prime Minister's Questions, Mr Robertson described the UK government's response as "frankly not good enough" and said it was a matter of emigration and not immigration.
The chancellor, standing in for Prime Minister David Cameron, said the family had not met immigration regulations and the home secretary would be writing to Mr Robertson on the matter.
The Brains, who have Scottish ancestry, moved to Scotland as part of an initiative backed by the Home Office and Scottish government to attract people to live and work in the Highlands and Islands.
However, the Home Office closed the scheme in 2012 and Mr and Mrs Brain were required to comply with different visa criteria.
Mrs Brain had a student visa and has been applying for jobs after completing a degree in Scottish history and archaeology. Her husband has been working, but a recent job offer was withdrawn after a change in circumstances at his prospective employer.
Mr Blackford, MP for Ross, Skye and Lochaber, said the couple have continued to strive to meet the work requirements of the rules.
But he told the BBC's Victoria Derbyshire programme that the family's case was a human rights issue.
He said: "Lachlan reads and writes in Gaelic and would be going to a country where the first language is English. I am asking the UK government to show some compassion."
In a statement to the programme, the government said the family had been given extra time to meet visa rules.
An Australian couple's deportation is a human rights issue because their seven-year-old son's first language is Gaelic, SNP MP Ian Blackford has said.