The Spanish forward already has two Scudetto titles from his time in Italy with Juventus between 2013 and 2015.
The 34-year-old started his career at Athletic Bilbao, making over 320 appearances and scoring more than 100 goals before his move to Italy.
After Juventus, Llorente then spent time with Sevilla and Swansea before his move to Spurs in 2017 for a reported £12 million.
At the north London side, he scored 13 goals in 66 appearances, including significant goals last season against Borussia Dortmund and Manchester City as Tottenham reached the final of the Champions League.
Llorente had been without a club since he left Spurs at the end of his contract in June.
Liverpool are set to offload youth team striker Bobby Duncan to Fiorentina after the Serie A side made an offer to buy him permanently.
The PA news agency understands the Italians will pay €2m up front with a 20-per-cent sell- on fee for the player.
Last week the club were firm in their stance that the 18-year-old, a scorer of 32 goals for the Under-18s in his first season since arriving from Manchester City, would not be leaving.
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