The 23-year-old flanker, who has nine caps for his country, is now four weeks into his recovery and hopes to return to action in November.
It means Botham, who was a regular for Wales this summer, is unlikely to be fit to start the autumn series matches with New Zealand, South Africa, Fiji and Australia.
Wales head coach Wayne Pivac will begin his preparations for a mammoth autumn series in the next few months.
From Cardiff’s perspective, Botham is the only new significant injury ahead of the 2021-22 season with Luke Scully and Mason Grady continuing their recoveries from knee ligament injuries.
Ben Murphy (knee) and Ben Thomas (groin) are both currently unavailable but are progressing through their rehabilitation programmes well and are on track to return during the opening rounds of the season.
Wales and British & Irish Lions duo Josh Adams and Josh Navidi are due to return to training in two weeks’ time.
The pair both featured for the Lions against South Africa this summer.
