The Swansea striker is on Antonio Conte’s radar as he looks to boost Chelsea’s title hopes.
Conte wants Llorente to ease the burden on Diego Costa.
Chelsea’s current back up striker Michy Batshuayi is yet to start in the Premier League.
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Fernando Llorente has scored six goals for Swansea so far this season
And Conte, having worked with Llorente at Juventus, believes the Spain forward has more to offer his side this season.
Swansea are loathed to lose Llorente, who has six goals for the relegation battlers.
But the move appears to be picking up steam and could be complete in the next few days.
The Daily Star report that talks are progressing and a conclusion is close.
Chelsea's Thibaut Courtois scores free-kick in training
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Conte’s side sit five points clear in the Premier League but had their winning streak stopped by Tottenham last week.
And the Italian is desperate to press on with January deals to help their title charge.
He has already recalled Nathan Ake from Bournemouth and is working on deals to replace Oscar and John Obi Mikel.