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Leeds have been involved in back-to-back 4-3 thrillers after returning to the English top flight, with a narrow defeat suffered at defending champions Liverpool before then edging out Fulham at Elland Road.
Bielsa’s side were famed for their defensive resilience when securing promotion out of the Championship, but those qualities have briefly deserted them.
The plan is, after missing out on the likes of Ben White, to bring in Llorente and start plugging those leaks.
Discussions regarding a move to England for the 27-year-old centre-half are at an advanced stage and a switch should be formalised pretty quickly.
Llorente does have a €50m (£46m/$59m) release clause in his contract and is tied to Sociedad through to 2022, but Leeds have agreed terms below that mark.
They will now be offering a Premier League stage to a man who has seen a move to England mooted for some time.
A model of consistency for the Basque club has taken in 88 appearances to date, scoring eight goals.
The Ligue 1 outfit are said to have put €15m (£14m/$18m) on the table, with Sociedad quickly making it clear that no agreement would be reached at that figure.
Leeds have had more luck when it comes to negotiating a fee and are now in a position to welcome another fresh face into their squad.