Gregg Brain revealed last night that his family had been given hope of a job offer which could keep them in the UK.
He said a “major Scottish company” had been in touch, leaving he and his wife “cautiously optimistic”.
They are now expecting a call from its chief executive after a representative of the firm rang asking for details of the visa requirements.
Mr Brain said his understanding of the Home Office’s position was that if a post could be secured quickly and a new application submitted, it would be considered.
He added: “We have not given up simply because the deadline has passed because we may still have an opportunity.”
Kathryn Brain, who has a degree in Scottish history and archaeology, is seeking a historian or curator role.