According to the Telegraph, the 32-year-old is the primary target for Antonio Conte as the Blues manager is looking to bolster his strike force for the 2017/18 season. His side are planning a £15m ($19.4m) bid for the Welsh club's attacker.
Diego Costa led the west London club's attack as they went on to win the title last term. The Brazil-born Spain international is not included in the former Juventus manager's plans at Stamford Bridge. Conte had earlier sent a text to Costa saying that he is no longer wanted at the club.
Chelsea have already signed Alvaro Morata from Real Madrid as he joins Michy Batshuayi as the two recognised players to leave the Blues' attack. Llorente is the player identified by Conte as the pair have already worked together when they were at Juventus.
The Spain international is aware of interest from the English champions and the striker is keen on reuniting with Conte at the capital club. The player is also aware that Chelsea are set to make an approach for his services.
Swansea, on the other hand, are determined to retain Llorente at the Liberty Stadium. However, Paul Clement's side are aware that the opportunity to play in the Champions League is expected to be the difference.
Conte wanted to bring Llorente to the Bridge in the January transfer window. However, Swansea rejected their approach as they were hoping to avoid relegation from the Premier League last season.
Clement and Swansea are braced for another approach from Chelsea for Llorente before the deadline day. The striker is currently on the sidelines after suffering an arm injury even before the start of the season and is yet to feature for the Welsh club.