The 31-year-old Frenchman has been linked with a move to Italian giants AC Milan.
A close family member told the Sun: “William is not a happy man at the moment.
“He did not apologise to his team-mates over his comments, nor did Arsene Wenger make him do so.
“But he still feels completely humiliated to have been stripped of the captaincy and is strongly thinking about a move in January.
“The summer would be too far away.
“He says he is lonely in the dressing room and at training, and he doesn’t really have many friends.
“William doesn’t want to return to France but AC Milan have publicly shown their interest in him and that would be an attractive option.”
The source added: “Yes our fans were very supportive at Chelsea and William was very grateful for that.
“But his body language on the pitch says everything you need to know. He’s definitely not the same man after the whole fiasco over the captaincy and the comments about his team-mates.
“He has a contract with the club but even a blind man knows that he is unhappy and could end that miserable time sooner rather than later.”