The 31-year-old, who was part of the Spain squads that won the 2010 World Cup and 2012 European Championship, is due to undergo a medical with the club on Friday.
It is understood the former Athletic Bilbao and Juventus forward has been handed a two-year deal by the Premier League side.
Llorente arrives at the Liberty Stadium with a considerable pedigree, having won five major trophies at Juve – including three Serie A titles – and appeared as a substitute in the 2015 Champions League final.
But he was not a regular starter by the time he returned to La Liga with Sevilla, where he largely acted as Kevin Gameiro’s understudy.
He started 14 times in his solitary season, scoring four times.
No transfer fee has been announced for the player, though Sevilla were keen to move him along having made him one of their highest earners when he arrived on a free transfer last summer.
Swansea were in need of attacking reinforcements, having allowed Alberto Paloschi, Bafetimbi Gomis and Portugal’s Euro 2016 hero Eder to move on.