The back-row forward, who can also play lock, has been ruled out for up to 12 weeks following the injury suffered in the 9-6 European Champions Cup win at Scarlets last Saturday.
It is a fresh setback for the 25-year-old after he was a surprise omission from Scotland’s World Cup squad.
Harley has been replaced by Ryan Wilson at blindside for Sunday’s 1872 Cup clash at BT Murrayfield and Glasgow head coach Gregor Townsend said: “Rob is going to be out for up to three months which is another blow for him because I think he’s been in great form.
“It was one of those accidental injuries, a real bad one – a dislocated thumb. He had to get surgery and to come back on the plane with us was great, but he’s going to be out for a few months so it’s a blow for him. It’s a blow for us too, but we’ve got really good players competing for that position. It was typical Rob – a bad injury and he went another two or three phases still playing till eventually he said to the physio ‘I might have a problem here’, with a bone sticking out a different way.
“It’s a shame. He got injured in the last two minutes of the 1872 Cup here [last season] and missed six weeks then went straight into a Six Nations game – France away. He loves this game as much as or even more than other players.
“The good news is that he was operated on down in Swansea. He knows the timescale on his recovery now and he was in good spirits yesterday but he’ll miss playing.”