LONDON (Reuters) - An Australian family facing deportation from the Scottish Highlands after a reversal in visa rules has been told to leave despite their pleas to stay, their local Member of Parliament (MP) said on Wednesday.

Gregg and Kathryn Brain, moved to Scotland from Brisbane in 2011 with their son Lachlan, now 7, are hoping for a job offer or a fourth extension to their visa after the last one expired at midnight on Monday.

The case has attracted much sympathy in Scotland, where Kathryn Brain was invited to study and then work as part of a British government-backed scheme to shore up the Highlands' aging and shrinking population.

That scheme was subsequently scrapped to address Britons' immigration concerns, leaving the Brains in limbo.

Ian Blackford, MP for the Dingwall area where the Brains live, said he had been told by the interior ministry (Home Office) on Tuesday night that the Brains should make arrangements to leave.

"I am astonished with the callous disregard to the interests of the Brain family," he wrote on his Facebook page, accusing the ministry of refusing to recognize the support of the local community and the devolved Scottish government.

Blackford said the government was "pandering to an anti-immigration mantra to drive the net UK migration numbers down" and said he would continue to fight the family's cause.

The case has attracted the attention of Scotland's First Minister Nicola Sturgeon, whose government has no say on immigration but who made a personal appeal on the family's behalf in May.

Kathryn Brain, who studied Scottish history and has been seeking work as a curator, was offered what appeared to be visa-compliant work at a Highlands distillery this year. But the offer was withdrawn in late July, giving the family little time to search for an alternative before an Aug. 1 deadline.

Gregg Brain, a health and safety expert, told Reuters this week that local support for the family had been overwhelming.

"We signed up for a deal and built a life here, only to have the ground pulled out from under our feet," he said.

On Tuesday, a Home Office spokesman said: "We have given (the Brains) three extensions on an exceptional basis over a number of months to allow them to try to secure a job that would allow them to meet the immigration rules, but this cannot be open-ended."

(Reporting by Stephen Addison; Editing by Louise Ireland)