The News of the World says Campbell is being lined up for a summer move by West Ham, Newcastle and Celtic.
The former England defender is now a free agent after seeing out his short-term deal with Arsenal which expired at the end of last season.
Campbell, 35, impressed on his return to Arsenal and showed he could still perform in the Premier League and even the Champions League.
Newcastle boss Chris Hughton knows Campbell from when they worked together at Tottenham almost a decade ago, while Avram Grant knows him from his first spell at Portsmouth before he moved on to manage Chelsea.
Celtic will need experience in defence if they make it into the Champions League and he has already described a move to the SPL as "a serious option."