The centre-back will arrive in London despite Paris Saint-Germain attempting to hijack the deal at the last minute.
PSG boss Thomas Tuchel wanted Silva to stay in Paris, but the club hierarchy were keen to move on the 35-year-old.
Sky Sports claim Silva’s relationship with PSG’s sporting director Leonardo has deteriorated and the Brazilian wants to move to Chelsea.
Silva will become Frank Lampard’s fourth summer signing and the ‘final touches’ on the deal to bring him to Chelsea have been agreed.
Chelsea have already signed Hakim Ziyech, Timo Werner and Xavier Mbuyamba and Silva could spark a flood of incomings at the London club.
Bayer Leverkusen star Kai Havertz is on the verge of joining Chelsea, while the Blues are also pushing to complete a deal for Leicester City left-back Ben Chilwell.
Silva played his final game for PSG in the Champions League final and won seven La Liga titles with the French side.
Lampard has targeted a massive overhaul of his squad this summer after managing to guide Chelsea to a top four place in the Premier League.