The operation - had been planned for some time - was a success but Thompson will need time out to recuperate and rehab. He'll miss Glasgow's pre-season fixtures against Worcester Warriors and Ulster as well as the opening URC fixtures.
Warriors have brought in experienced stand-off Joel Hodgson on a short-term deal to provide back-up to Duncan Weir and Domingo Miotti.
Prop Enrique Pieretto injured his shoulder playing with the second-string Argentina XV in the summer and will be out until November following surgery.
Lucio Sordoni has joined Warriors on a short-term contract to cover for his fellow Argentinian international.