The South Korean enjoyed a consistent season at the Reebok Stadium after missing much of the previous campaign with a broken leg.
New Everton manager Roberto Martinez is thought to be looking at the 24-year-old as he prepares to re-shape the squad he inherited from David Moyes.
Paolo Di Canio has already made several new signings since the end of last season, and looks intent on bringing more new faces to the Stadium of Light.
Liverpool and Stoke City were rumoured to be keen on Lee in the January transfer window but no formal bid was made for the winger.
Bolton went close to edging into the play-off places last term, but having just failed to make the cut it remains to be seen if Lee will play in the Championship again this season.