Blues boss Antonio Conte had been linked with a swoop to sign the 32-year-old former Spain international during the January transfer window, but the Swans rebuffed their advances because they needed him in their fight to avoid relegation.
Llorente bagged 15 goals for the South Wales outfit and ultimately helped guide them to safety, while Swans boss Paul Clement sees him as a key part of his side looking ahead to next season.
The former Athletic Bilbao man recently suffered a broke arm on holiday but should be fit for the start of the new Premier League season next month so that has not put off the Blues.
Llorente was at Juventus during Conte’s time in charge in Turin so the Italian knows all about his quality, and as the player is now inside the final year of his current contract at the Liberty Stadium the Blues could offer the Swans a sizeable fee to tempt them into a sale this summer.
It seems Conte has switched his sights back to Llorente because his first-choice target Lukaku appears to be leaning towards a move to Manchester United from Everton in a potential record £100million deal.