Manager Neil Warnock said on Friday that Becchio, United's 19-goal leading scorer, had rejected a new three-and-a-half "Premier League" contract and had asked to leave after a number of bids from overseas clubs were turned down.
But according to Becchio`s agent, Horacio Rossi, the efforts to keep Becchio by GFH Capital, the club`s new owners, are not enough. Speaking to Press Association Sport, Rossi said: "Luciano is on a spectacular run of form at the moment and for that reason Leeds came to us at the moment that Europa League clubs declared their interest in Luciano."
He added: "As the club had declared that they would not accept any offer in this transfer window, I proposed to them the idea of extending Luciano's contract with Leeds until 2017 so that Luciano could end his career at the club.
"But for a new contract, there has to be an improvement. The offer that they made to us was insufficient. We respect the club's efforts, but it isn't enough. Leeds is like a home for Luciano. He feels respect and love for the club's fans. He has the best intentions of staying. He was willing to give up a lot of money to stay at Leeds."
Leeds said in a statement this week that Becchio was already their highest-paid player, but Rossi has also cast doubt on that. He said: "Luciano has to think about his family. He has three children. He has to look for security for them. He is 29-years-old, not 25. He is only a footballer once.
"Leeds say that Luciano is the most expensive player in their squad. They said on their website that he is the player who earns most but I don't know if this is true. It doesn't seem right to me. I don't know what (El-Hadji) Diouf, or (Ross) McCormack earn."
Rossi did praise Warnock for the "excellent" way in which he has handled the situation, but did add that he felt GFH Capital could have done more.
"Neil is a footballing man, a successful man and he has understood the situation well," Rossi said. "It is my opinion that the new administration of Leeds United has not put in enough effort in their attempts to retain Luciano."
Warnock said on Friday he would see if the player`s head was right to be involved for Sunday`s fourth-round tie. Press Association Sport asked GFH Capital to comment but had yet to receive one.