Swansea will not, however, sell Llorente to Chelsea unless they can sign a replacement but they are keen on bringing in Wilfried Bony to join Tammy Abraham, who is on loan from the Blues.
Llorente was a Chelsea target in last January's winter transfer market but Swansea manager Paul Clement blocked the sale as he made the former Spain international integral to the club's bid to survive relegation.
Chelsea haven't yet made the significant breakthrough required for the former World Cup winner, who has played alongside Alvaro Morata at Juventus, as well as being managed by Antonio Conte in his two-year spell at the Turin club.
Alex Oxlade-Chamberlain remains Chelsea's closest deal to completion after a £35m fee was agreed yesterday , while they are also pushing to finalise a similarly-priced deal for Leicester City's Danny Drinkwater.
Diego Costa, meanwhile, is pushing for a move to Atletico Madrid and all parties are keen to make it happen. Chelsea's asking price for the Spain international remains £40m as they look to get the market value for the striker who remains in Brazil with his family and friends.