The Toffees have endured a difficult summer in terms of recruiting players, with Maarten Stekeleburg and Idrissa Gueye the most notable additions at Goodison Park.
A number of big name targets have evaded their grasp, but the club remain keen to recruit, with reports claiming Sporting CP’s Patricio is a target.
However, the Merseysiders will only move for the Portugal No.1 if Robles follows Tim Howard out of the exit door.
It’s understood that the 28-year-old wowed Koeman as he starred for his nation at the European Championship and is seen as a viable first choice stopper by the Dutchman.
Robles was Everton’s main goalie for much of the end of last season with the Spaniard having moved ahead of Howard amid question marks surrounding his ability.
Stekeleburg has been signed, but the ex-Fulham man’s career has somewhat stagnated in recent years.