Swansea are working on securing a replacement for Llorente as they soften their stance on keeping the experienced striker, but Mauricio Pochettino's side believe they have a chance in stealing a march on their rivals.
Antonio Conte has been keen on a reunion with the former Spain international since the January transfer window, but Paul Clement blocked the move as the striker became pivotal in the club's survival in a relegation battle.
Llorente played under Conte at Juventus and also worked alongside Alvaro Morata in his two year spell at the Italian giants, but it remains to be seen which London club he will chose to sign for.
Chelsea will be keen to get this deal over the line after failing to sign Alex Oxlade-Chamberlain, who is on the verge of signing for Liverpool in a £35m deal after rejecting the Blues as they attempted to open contract talks.
The Premier League champions also remain in the hunt for Leicester City's Danny Drinkwater, who has handed in a transfer request at the King Power Stadium in a bid to force through a move.
Diego Costa is still expected to leave Stamford Bridge for Atletico Madrid but no agreement has been found as the transfer window closes tomorrow at 23:00 BST.