The striker is yet to feature for the Swans this season after breaking his arm while on holiday with his family, but has repeatedly been linked with a switch to the Blues.
Chelsea boss Antonio Conte signed Llorente for Juventus in 2013 and is believed to see him as the ideal candidate to add depth to his forward line.
The 32-year-old only joined the Swans last summer and played a key role in their successful survival bid under Paul Clement.
However, he now looks set to bring the curtain down on his stay in south Wales after the two clubs reportedly agreed a deal worth £15 million for his services.
Swansea have already sold Gylfi Sigurdsson this summer and boss Clement looks set to be a busy man in the final 36 hours of the transfer window.
If the deal goes through, it is reported that the Liberty Stadium outfit will attempt to sign former striker Wilfried Bony from Manchester City, having sold the Ivorian to the Citizens in January 2015.