Francis hurt his shoulder while playing in the Rugby World Cup semi-final defeat against eventual champion South Africa.
Exeter Chiefs confirmed the 27 year-old will miss up to four months of action due to a "serious" problem.
That provides an additional selection headache to new Wales coach Wayne Pivac, with their Six Nations campaign to start on February 1 against Italy and their last match against Scotland taking place on 14 March.
Wales is already without star centre Jonathan Davies for the entire tournament, with fly-half Rhys Patchell also a doubt.
As for Exeter, the Premiership club have signed Argentine Enrique Pieretto due to the injuries to Francis and fellow prop Greg Holmes.
Director of rugby Rob Baxter said: "After two or three scans it became clear that Tom needs an operation to repair the shoulder injury he picked up with Wales at the World Cup.
"It's not great news for us, but at least we know where we stand now."