Kellock's contribution in the 9-7 reverse to the Blues at Scotstoun was ended prematurely and he wasn't available for selection for last weekend's postponed clash with Treviso.
The 32-year-old has since had surgery and isn't expected to be fit again until late April, an absence that will rule him out of the final two pool games in this season's Heineken Cup as Glasgow head to Exeter before hosting reigning champions Toulon.
He will also miss Scotland's entire RBS 6 Nations campaign and will have to wait until the summer to add to his tally of 56 international caps.
"Gutted about the arm but part of the job. I've had a good run for the last few years - back soon enough," Kellock wrote on his Twitter account.
Glasgow head coach Gregor Townsend admits that losing one of his most experienced players is a badly timed blow given that his side face Edinburgh in back-to-back domestic derbies over the festive period.
"He has played a prominent role in these games over the last few season and thrives against Edinburgh. We wish him well in his recovery and he still has a big role to play at the club as he works his way back to full fitness," added Townsend.
Kellock has played 132 times for Glasgow since making his debut in 2006 and has featured in European competition on 58 occasions for the Warriors and previous club Edinburgh. He made his 50th Heineken Cup appearance in this season's Round 2 win over Exeter and has started all four games in the tournament to date.
Glasgow currently sit bottom of Pool 2 after being beaten home and away by the Blues, having previously seen off the Chiefs at Scotstoun and picked up a try-scoring bonus point in defeat at Toulon.