The 27-year-old had the operation to fix a recurring shoulder problem.
Glasgow Warriors head coach Gregor Townsend said: "We're disappointed for Jon that he's going to miss the first few weeks of the new season.
"He played a lot of rugby for us last season and he worked hard to become a consistent performer.
"His recent surgery will allow him to train fully on his return and play even better next season. We wish him all the best in his recovery period."
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