Llorente, who is considered one of the more technically gifted target men in La Liga, has struggled for regular game-time during his spells with the Europa League Champions and Juventus over the past two seasons, and is said to be lined up as the Swans marquee summer signing.
"[Llorente] is one of the players who has been mentioned," first-team coach Alan Curtis told BBC Football. "It's probably time we brought a big signing in."
Swansea have also enquired about Leicester City's Leonardo Ulloa as they look to secure a target man, having loaned Bafetimbi Gomis to Ligue 1 side Marseille for the season.
The 31-year-old Llorente made his name at Athletic Bilbao, scoring 111 goals and registering 36 assists in 327 games during his nine seasons with the club.