Llorente turned down the offer of a new deal at the Primera Liga club over the summer and said he wanted to leave, but Bilbao refused to negotiate with interested sides.
The Spain international, who has previously been linked with Juventus and Arsenal, has found himself relegated to the bench so far this season, starting one league game and scoring a single goal.
After failing to persuade him to stay, Bilbao could chose to sell the 27-year-old in January rather than lose him on a free transfer in the summer, with Juve thought to be planning a winter swoop for the striker.
He said: "I am going to leave on June 30. I want to develop as a player and try something new."
Llorente, who made his senior debut with Bilbao in January 2005 after coming through their youth academy, was the club's top scorer last season with 17 goals in 32 league appearances.
The uncertainty over his future has made him a target of abuse from some sections of Bilbao's fans and he added: "It isn't about money because the offer the club made was irresistible.
"The media have not helped with the negotiations and it is one of the things that has generated the bad atmosphere around me out on the pitch. This has encouraged me not to renew my contract; this and many other things.
"I took the decision when I went out for the first Europa League game at San Mames (in August).
"Most of what has come out in the press is lies and this has turned people against me. This has meant when I run out on to the pitch people whistle at me. There is a sector of the crowd who do not want me but I feel loved by the majority."