The right-back looks likely to be denied the chance to face former club West Brom on the opening day of the new campaign next Saturday as he battles to return to full fitness.
Jones joined Sunderland from Albion on a free transfer in May following a three-year spell at The Hawthorns, but will have to wait to make his competitive debut.
"I think it will take at least another week for him to train with us," Sunderland head coach Gus Poyet told The Chronicle.
"We're managing the situation the best we can."
Jones will join Emanuele Giaccherini in watching on from the sidelines against West Brom after the winger was also ruled out of next weekend's game with a groin problem.
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