The Spaniard left the club when his contract expired at the end of June, but Spurs have offered the former £100,000-per-week attacker a new one-year deal on reduced terms.
However, they are yet to hear back from Llorente and are beginning to lose hope of him returning.
The 34-year-old was linked with Manchester United earlier in the summer and is now attracting interest from a host of Serie A clubs.
Llorente scored 13 goals in 66 games during his two seasons at Tottenham.
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