The negotiations are centered around a potential transfer fee, but no agreement has been reached yet.
Everton has been looking for a new goalkeeper after losing Joao Virginia and Asmir Begovic, whose contracts expired last month.
The club sees Travers as a good backup option for their first-choice goalkeeper, Jordan Pickford.
Travers, a Republic of Ireland international, spent part of last season on loan with Middlesbrough.
Bournemouth is open to selling Travers as they are engaged in talks with Chelsea over a £25m deal for another player, Djordje Petrovic.
