White Hart Lane and Emirates Stadium sides battle for experienced marksman.
The 30-year-old is set to leave Juventus after the Turin side brought in Paulo Dybala and Mario Mandzukic from Palermo and Atletico Madrid respectively.
The Serie A side saw Carlos Tevez leave for Boca Juniors but can also call upon Spanish forward Alvaro Morata, leaving World Cup winner Llorente out in the cold.
It would take a fee of £8.5m to bring in the former Athletic Bilbao man, who has netted 23 goals in 65 Serie A appearances.
Prior to his move to Italy the lanky Spanish international bagged 118 goals for Athletic Bilbao and could offer either Arsene Wenger or Mauricio Pochettino a strong option in the final third.
Fernando Llorente has the physique to thrive in the Premier League and has the experience to prove a very decent signing given the relatively low asking price.