Llorente has been heavily linked with a move away from Bilbao for some time after turning down the offer of a contract extension over the summer.
Reports over the weekend suggested Llorente has his heart set on a move to the Premier League despite interest from Juventus, and held a meeting with Manchester City at the tail end of last season over a potential switch.
Now though Arsenal appear to be finalising their January plans and have been linked with a move for Llorente, who the Gunners reportedly see as the ideal long-term replacement for Robin van Persie.
Arsenal manager Arsene Wenger is said to have a sizeable transfer kitty at his disposal according to news reports last week, and should be able to make an acceptable offer to Bilbao for their star striker.
The Spaniard is out of contract at the end of the season and could be available for around £15 million.
Recently Wenger has opted to deploy Gervinho in a central striker role, away from his usual position out wide, while forward Olivier Giroud has dropped to the bench as he continues to adjust to life in the Premier League.
Lukas Podolski arrived at the Emirates Stadium over the summer but is more of a wide player, while Theo Walcott is yet to be given the opportunity to prove he can play as a striker as he claims.
There appears to be room in Arsenal’s squad for another striker - although if he wants to land Llorente he must first fight off competition from other Premier League clubs, including rivals Tottenham.
Spurs have been linked with a move for the 27-year-old in the past and could try for his signature in January, given their lack of firepower up front.
Spurs can only call upon Jermain Defoe and Emmanuel Adebayor to lead the line despite the arrival of Clint Dempsey over the summer.
Spurs seemingly have money to spend in January and boss Andre Villas-Boas may prioritise bringing in another forward.
Reports in Spain yesterday suggested it could be a straight shoot-out between Arsenal, Tottenham and Manchester City – although the Citizens appear well stocked in the forward department unless one most likely Edin Dzeko or Mario Balotelli, moves on.