Edinburgh’s Matt Scott has been called into the Scotland squad for the tour to the far east. The first Test is on 18 June.
A scan revealed that 26-year-old Dunbar sustained a bone injury in the 13th minute of Glasgow’s final regular-season Pro12 match - also against Connacht in Galway.
He will not need surgery and a Warriors’statement said he will now begin “a period of rest and recovery” with the club medical team.
Glasgow head coach Gregor Townsend said: “We’re disappointed for Alex that his season has been cut short through injury.
“He had been playing very well and played a big role in getting us into the Guinness Pro12 play-offs.
“We wish him all the best with his rehab and we look forward to him returning to training ahead of the new season.”
Edinburgh’s Matt Scott has now been called up by Scotland coach Vern Cotter for the Japan tour.