Marca says the 33-year-old is the 15th highest goalscorer in Athletic history on 118 goals, just two behind Fidel Uriarte.
Last season, Real Sociedad had pondered a move for his services, with Llorente married to a native of San Sebastian and he spent most of last summer in Donostia, fuelling rumours about his future.
He would be open to leaving Tottenham for a project that would suit him and given his affinity with Athletic, they would have preferential treatment in any conversation.
Spurs would not stand in his way if he were to seek an exit during the summer.
"It is my home and wonderful things happened for me there that will never be forgotten," Llorente said last summer when quizzed on a return.
"Athletic will always mean so much to me. If they were to propose a comeback, I would certainly think about it."