The Boca Juniors youth product has scored 36 goals in two seasons since arriving from Spanish club Merida, and helped Leeds win promotion back to the Championship in May.
The Daily Mail report that the club approached Becchio about a new deal before Christmas and he is known to be keen to stay at Elland Road.
But despite a series of positive meets, the 26-year-old has now been told there will be no extension to a contract that runs out next summer.