Edinburgh back rower McInally will fly out from the UK later today as Scotland prepare for Tuesday's clash with the Wallabies in Newcastle, New South Wales.
The 21-year-old has captained his country ay U17, U18 and U20s and has also represented Scotland at Sevens and A level.
He missed Edinburgh's Heineken Cup semi-final defeat to Ulster with a hamstring injury but he did feature in four of his side's pool games and the historic quarter-final win over Toulouse.
His inclusion is a result of a hand injury sustained yesterday by John Barclay who is now being monitored by the Scotland medical staff which includes veteran Lions doctor James Robson.
Glasgow centre Dunbar was in outstanding form for his club at the back end of the season and has been rewarded for a string of fine performances due to the fact that Max Evans is still involved in Top 14 action with Castres.
Evans is set to feature in this weekend's French Championship semi-final with Toulouse after his side saw off Montpellier in the first round of play-offs last week.
Dunbar is currently visiting family in Tasmania and will therefore join up with the party in Sydney tomorrow.
"We've called up Stuart as a precaution given a hand injury that John Barclay sustained in training yesterday," said Robinson.
"John should be available for selection for the Test but we felt it better to ensure we have sufficient back-row options.
"Alex will meet up with the squad tomorrow. We've gone for him as we are still liaising with the IRB, French Federation and Castres to have Max Evans join us as soon as possible."
After facing the Lions' next opponents on June 5, Scotland take on Fiji on June 16 before ending their tour in Samoa on June 23.