The custodian has been trailed by a number of clubs after Chelsea decided to release him at the end of last season.
Evans has already rejected moves to both Swindon and Southend, but it now appears The Trotters are ready to offer him the chance to resurrect his career in the top flight.
Bolton boss Sam Allardyce is keen to bring in back-up for first choice Jussi Jaaskelainen, and confirmed he would be eager to take Evans to The Reebok Stadium.
"Rhys is a player who interests us," declared Allardyce.
