But the competition to sign the Uruguayan is tough with Premier League sides Watford and Aston Villa also in the running for the 22-year-old.
It had been suggested that the powerful 6ft 5in striker was close to sealing a switch to Serie B side Cagliari but, according to the Corriere dello Sport, Avenatti's next destination is likely to be in England.
Leeds scouted the former River Plate hitman on numerous occasions last term, in which he struck 11 goals for the second-tier Italian outfit.
Avenatti is determined to prove himself at the highest level and is said to be keen to move on this summer, however Ternana will not allow their star man to leave for less than £5m.