Llorente, 32, joined Swansea on a free transfer last summer and notched up 15 league goals as Paul Clement's side just about achieved survival. The former Juventus and Sevilla hitman remains a key part of Clement's plans but has not featured for the south Wales outfit so far this season after suffering a broken arm in a cycling accident while on holiday.
Everton manager Ronald Koeman is determined to land a striker of high calibre before the end of the transfer window and has identified Llorente as a target, though the Toffees are facing strong competition from Premier League champions Chelsea, according to ESPN.
Chelsea were interested in Llorente during the January transfer window and are keen to bolster their striking options in the absence of Diego Costa, who will almost certainly never play for the club again. The fiery Spain international is determined to return to Atletico Madrid, but Everton boss Koeman admitted recently that he would "welcome" the 28-year-old at Goodison Park.
Everton parted with £45m to sign Gylfi Sigurdsson from Swansea a couple of weeks ago and could well reunite the Icelandic international with his former attacking accomplice as they look for a fitting replacement for Romelu Lukaku, who joined Manchester United earlier in the transfer window.
The lure of playing for Chelsea and living in London may play a part in Llorente's decision over where to ply his trade next but his chances of game-time would be substantially higher at Everton, who recently sent scouts to run the rule over Benfica forward Raul Jimenez.
Llorente is not the only towering forward to be linked with a move to Everton; the Toffees enquired over the availability of Cardiff City forward Zohore, who has flourished under the tutelage of Neil Warnock in recent months, but the Bluebirds quickly rebuffed the Merseysiders' approach and have no interest in parting with the £30million-rated Dane, according to BBC Sport.