Chelsea boss Antonio Conte has been keen on Llorente for several months and tried to sign him in January.
The Blues need to bolster their attacking options with Diego Costa still ousted and the club are expected to try and land Llorente before Thursday’s transfer deadline.
But Conte and co will now reportedly face competition from Premier League rivals Everton.
ESPN claim the Toffees have made Llorente one of their main targets for the remaining days of the summer window.
Ronald Koeman wants to add another striker to his ranks following Romelu Lukaku’s £75m move to Manchester United.
And Everton and Chelsea now look set to go head-to-head for Llorente’s signature.
Swansea are reluctant to let their Spaniard leave, especially having already lost Sigurdsson to Everton earlier this month.
But ESPN add the Swans could be tempted to sell if they receive a hefty offer for the 32-year-old.
Llorente was hugely impressive for Swansea last season, scoring 15 Premier League goals.
He is yet to feature for Paul Clement’s side this term after breaking his arm during the summer.
The ex-Sevilla star is expected to be fit again after this weekend’s international break but it remains to be seen who he will be playing for.
