Per The Times' Gary Jacob, the Blues initially pursued the 32-year-old Spaniard back in January, but he was not allowed to leave Swansea.
He only has one year remaining on his current deal at the Liberty Stadium, but the Welsh outfit are "determined" to keep hold of him, Jacob added.
However, Chelsea's first-choice No. 9 Diego Costa is reportedly nearing the Stamford Bridge exit door and is on the verge of returning to Atletico Madrid, per Spanish football writer Manu Sainz (via The Sun's Alex Terrell).
As a result, the Blues need added firepower, and Llorente would be a sensible and useful addition even if Costa remains in west London, as Chelsea need a bigger squad for next season when they return to the UEFA Champions League.
Per WhoScored.com, Llorente was clinical as he netted 15 goals and provided one assist in 33 Premier League appearances last term to help Swansea avoid relegation: