The arrival of Real Sociedad player Llorente, 27, would provide another senior centre-half option for United, who brought in Freiburg centre-back Robin Koch during the close season.
Following Ben White’s return to Brighton and the south coast club consistently rejecting the overtures of Leeds to sign him permanently, the Elland Road outfit have been keen to bring in another experienced defensive option into their back four – alongside Koch.
As it stands, Leeds – who have conceded seven goals in their opening two Premier League games – have inexperienced back up in the likes of Pascal Struijk and any injury problems in defence would leave them significantly short.
Reports suggest that Leeds could complete the signing of Llorente by the weekend.
Leeds have switched their attentions to Llorente following Dinamo Zagreb defender Josko Gvardiol’s decision to head to RB Leipzig despite strong interest from United.
Leeds remain in the market for several other players, including Udinese’s Rodrigo De Paul and Ukrainian midfielder Ruslan Malinovskyi, who plays for Atalanta.