West Brom have told Bakary Sako to lower his wage demands if he wants to move to The Hawthorns.
Highly-rated Atletico Madrid midfielder Saul Niguez has signed a new five-year contract.
Becchio, 29, the club's leading scorer this season with 19 goals from 31 appearances, has been the subject of much speculation during the transfer window and Leeds have confirmed they have received two overseas offers for the Argentinian.
"Luciano Becchio, who has been the subject of two separate bids from foreign clubs during the January transfer window, has handed in a transfer request to the club, as financially he believes his best interests are served by a move away from Leeds United," said a statement released by the club.
"Both bids have been rejected and the player will only be allowed to leave in the event that an acceptable offer is received.
"The club have sought to agree the terms of a new contract with the player, who is already the highest earner, but the requested terms were beyond a level that we could support, particularly in view of the regulations relating to Financial Fair Play that will be effective in coming seasons.
"It is disappointing that this situation has occurred when he is under contract until 30 June 2014."
Manager Neil Warnock last week expressed his disappointment in Becchio's performances in recent games, claiming the player had been distracted after having his "head turned" by agents.
Becchio arrived at Elland Road from Spanish club Merida in August 2008 when former boss Simon Grayson was in charge.
He scored 19 goals in his first season when Leeds were in npower League One, 17 the following season when the club won promotion and 20 in his first campaign in the Championship. Becchio has scored a total of 86 goals for Leeds in 221 appearances.
Leeds have this week been linked with a move for former Portsmouth striker Aruna Dindane, who recently left Qatar-based Al-Sailiya Sport Club.