The west London side have already made four additions in the summer transfer window following their Premier League title success last summer.
Antonio Conte has already signed Real Madrid striker Alvaro Morata in a £58m deal earlier this summer to improve his attacking options.
The Chelsea boss has told controversial centre-forward Diego Costa that he is free to leave the defending champions in the summer transfer window.
The Italian coach is thought to be interested in signing a striker to further improve his squad options ahead of a return to the Champions League.
Conte was linked with a move to sign Llorente in the January transfer window but a deal failed to emerge for the 32-year-old striker.
However, the title holders are thought to be ready to make a fresh approach to recruit Llorente before transfer deadline day.
Sky Sports News reporter Solhekol confirmed the Premier League champions want to sign the Spain international.
“Chelsea also still want to sign a back-up striker and they remain interested in Fernando Llorente,” Solhekol told Sky Sports News on Monday morning.
“He scored 15 times for Swansea last season. Antonio Conte knows him very well from their time together at Juventus.
“This deal may depend on Swansea signing a replacement striker. They’re still in talks with Manchester City about Wilfried Bony.”
Conte worked with Llorente during his spell in charge at Serie A giants Juventus.
The Blues were 2-0 winners against Everton at Stamford Bridge in the Premier League on Sunday.
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