36-year-old Silva is close to agreeing a one-year extension, with the two parties hashing out the final details of his new contract.
The Blues will then begin discussions with N'Golo Kante and Jorginho over new contracts, according to Fabrizio Romano.
Kante, 29, will become a free agent in 2023 and has become a target for Inter Milan.
Jorginho's deal expires at the same time as Kante, with the Italy international continuing to be linked with a return to Serie A.
Both players have thrived under Thomas Tuchel and the club wants to ensure they remain part of their long-term future.
Thiago Silva will stay at Chelsea - he wants to continue at #CFC, last details to be discussed... then his contract-extension clause until June 2022 will be triggered in the next weeks.
Contract negotiations with Kanté and Jorginho will start in the next months - no rush. https://t.co/6zBLSSlkZ0
— Fabrizio Romano (@FabrizioRomano) May 11, 2021