The 22-year-old left Manchester United for the La Liga club in the summer in a bid to rebuild his career after falling out of favour at Old Trafford.
However, Sociedad announced that Januzaj felt severe pain in his knee following Thursday's 3-1 defeat at Zenit St Petersburg in the Europa League and have not set a return date.
An official statement read: "Adnan Januzaj completed the match against Zenit St Petersburg without any complaints or apparent trauma, but an hour later began feeling intense pain in his right knee.
"The MRI scan taken yesterday showed a moderate articular effusion, which may be in relation to a previous meniscus operation.
"The initial plan will be to rest and undergo mild physiotherapy. His return to action will be progressive and will depend on how his recovery improves."