The 27-year-old was part of Spain's Euro 2012-winning side over the summer, although he did not play a single minute for Vicente del Bosque's team, and it is believed that he is keen to leave Los Leones and challenge for major club honours
The Bianconeri have spent the summer attempting to lure Arsenal's Robin Van Persie to Turin, but with further reports stating that the Dutchman has informed the Old Lady that he prefers a move to Manchester United, they appear to have switched their attentions to more attainable targets.
However, Arsenal may themselves now join the hunt for the Basque target man, should the sale of Van Persie materialise.
Elsewhere, it was reported by Voz Populi that Llorente had fallen out with Athletic coach Marcelo Bielsa earlier in the summer, and will not sign a new deal while the Argentine remains in charge at San Mames.
Bielsa reportedly believes that Llorente holds the coach responsible for his failure to feature for La Roja this summer.