The 26-year-old Spanish international has spent his entire career at San Mames, but with just 18 months remaining on his existing deal, there is a cloud of uncertainty surrounding his future according to skysports.com.
Spurs boss Harry Redknapp is keen to make a 'big signing' in January as he looks to keep the White Hart Lane club's title ambitions alive heading into the New Year, and Llorente may well be that man.
However, Tottenham could face competition from Liverpool amid increasing pressure on Reds boss Kenny Dalglish to add a goalscoring striker to his forward ranks at Anfield.
Meanwhile, Chelsea manager Andre Villas-Boas has also been credited with an interest in Llorente - who is believed to have been identified by the Blues as a potential replacement for the outgoing Nicolas Anelka - who will leave Stamford Bridge in January.
But any plans the trio of Barclays Premier League clubs may have had of signing the Spaniard have now suffered a setback, after Bilbao sporting director Jose Maria Amorrortu revealed.
"We are very confident that Llorente stays with us. He is a very important player. We are working on its continuity, but we have not gone any further.
"I'm optimistic. Fernando has been here for 11 years. This club is a club of values and Fernando knows this.
"We must transmit values and reinforce them because they are what make us different."