Llorente struggled to keep a regular place in the starting line up for Sevilla last season, scoring just four league goals, as well two in their successful retention of the UEFA Europa League.
The 31-year-old enjoyed netting double figures in five consecutive years with Athletic Bilbao in La Liga, before joining Juventus. But after a difficult second season, he moved back to his home country and with Sevilla.
But now Everton are hoping to bring him to the Premier League, as Sevilla are considering cashing in on the forward.
Ronald Koeman is building his new Toffees squad, with rumours constantly circulating doubting whether Romelu Lukaku will remain at the club. But even if the Belgian stays, the Dutchman still wants to add to his team, and improve the firepower that saw the club finish in the bottom half of the table last season.