The Express & Star says Jones today signed for Sunderland on a four-year deal after turning down a final contract offer from Albion.
Jones' decision mean Albion must replace both of last season's senior right-backs after they released Steven Reid.
Director of football administration Richard Garlick said: "Following further talks with Billy's representative, we made our final offer last week - a long-term deal on vastly-improved terms based on Billy's contribution over the past three years.
"Billy rejected this and has exercised his right to run down his contract with a view to joining Sunderland.
"We're obviously disappointed by Billy's decision but wish him well in his future career.
"With there being no guarantees that Billy would re-sign, we have been preparing for this scenario and will pursue the options we have been exploring in the right-back position."
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