The halfback, who has 31 caps, was released from the squad on Thursday and the Welsh Rugby Union confirmed he will undergo further assessments "to establish the best course of action".
Webb's setback leaves head coach Warren Gatland potentially without nine senior players ahead of the opening clash at home to Scotland on February 3.
Webb has been replaced by uncapped Cardiff playmaker Tomos Williams but Gatland will be missing a host of experienced names for most, if not all, of the Six Nations.
British and Irish Lions stars Sam Warburton, Dan Lydiate and Jonathan Davies have all been ruled out of the entire championship while Jake Ball's shoulder injury is likely to leave the lock sidelined for all the fixtures.
Taulupe Faletau's knee injury is set to keep the Bath No.8 out for the first three matches while five-eighth Dan Biggar is sidelined for a similar period due to a shoulder problem.
Fellow No.10 Rhys Priestland is currently out due to a hamstring injury and there are also doubts over fullback Liam Williams, who could yet require surgery on an abdominal injury.
Wales, who last won the Six Nations in 2013, travel to Twickenham to take on England on February 10 and then meet Ireland in Dublin on February 24.
The injury means Webb, who will join Toulon at the end of the season, could have played his last game for Wales due to the WRU's "senior player selection policy".
The governing body stated in October 2017 that "no player with less (than 60) caps will be selected for Wales if they ply their trade outside of the country".