By Elisabeth O'Leary

EDINBURGH (Reuters) - An Australian family living in the Scottish Highlands face deportation on Monday if a high-profile bid to comply with UK visa requirements fails.

Gregg and Kathryn Brain are hoping for a last minute job offer or a British interior ministry extension allowing them to stay, with their seven-year-old son Lachlan, after a deadline on their visa extension runs out at midnight.

The Brains moved from Brisbane to Scotland in 2011 on Mrs Brain's temporary student visa, which then allowed her to seek work afterwards as part of a British government-backed scheme to help shore up an ageing and shrinking population in the Highlands.

Visa rules were then changed, however, with an eye to addressing voter concerns on immigration in the UK as a whole. Immigration was at the centre of Britain's June vote to leave the European Union.

"One option is that there is mercy and compassion, and the Home Office (interior ministry) decides to offer an extension or find another way that they can stay," Scottish lawmaker Kate Forbes, who is helping the Brains, told Reuters.

The family's campaign to stay in Scotland, where their son has learned the ancient Scots language Gaelic, led them to a meeting with the devolved Scottish government First Minister Nicola Sturgeon, who personally appealed to the British government on their behalf in May.

An offer of work at a local distillery for Kathryn Brain, who studied Scottish history, was withdrawn last month after it became unclear whether it fulfilled immigration and employment regulations.

The legal complexities of the visa system meant more time was needed for job offers to be vetted for their suitability in resolving their predicament, Forbes said.

"The fear has always been that to grant too many extensions to the Brains would set a precedent. But we know there were few, if any, other individuals in the same boat as the Brains because the scheme was scrapped in 2012," Forbes said.

Having run down their savings as well as donations from well-wishers while trying to find a solution, the family is now surviving on the charity of neighbours, Gregg Brain, a health and safety expert, told the BBC.

The Brains argue that their case is exceptional because the rules were changed after they arrived.

The Home Office noted the family had already been given three grace periods of temporary leave to remain.

A spokeswoman for Prime Minister Theresa May said :"We recognise the strength of feeling on this case but there is a need to follow the rules, follow the process, and to date they have not lodged a visa application with the (interior ministry)."

If they are sent back to Australia, Gregg Brain said, the family will have to work hard to pay off debts accumulated while trying to resolve their situation.

"Brisbane is not Mogadishu ... but we would be going back to homelessness, jobless and indebtedness," he added. "It would take us the best part of 10 years to pay off our debts."

(Additional reporting by William James; Editing by Stephen Addison)