Gregg and Kathryn Brain from Australia say they have made contact with two employers who may be able to offer Kathryn a position which fulfils the requirements for a visa.
They admit all options, including voluntary departure, remain on the table after their leave to remain expired on Monday, which means they are currently living in the UK illegally.
Speaking to Premier Christian Radio, Gregg said he believes God has called his family to the UK and he has brought them to this point.
He said: "It has been our experience that when we feel these sorts of callings that God has opened doors for us, not just in the eleventh hour but well and truly into the 59th minute.
"Now this time, it seems that is had actually gone a little past midnight but these things seem to happen for us in this fashion just to remind us that it wasn't us who did it."
Gregg and Kathryn moved to the Highlands with their son Lachlan, now seven, in 2011 on Mrs Brain's student visa.
The government later withdrew a two-year post-study visa scheme and the Brains were given until Monday to meet the requirements for a UK tier two visa.
Gregg told Premier both he and his wife have Scottish heritage, adding: "We feel very much like we've come home; we feel a very deep connection with this area of the country.
"If we were able to stay, it would be the culmination of a calling that we have felt since we first came here on holiday in 2001."
The Home Office confirmed on Wednesday the Brain family's leave to remain would not be extended again and encouraged the family to leave the UK voluntarily.
Gregg Brain urged Christians to continue to pray for a "common sense and just resolution" from the Home office, and a job offer to stay.
You can listen to Gregg Brain speaking with Premier's Alex Williams by clicking here.