The PSG captain's contract at the French club ran out at the end of the previous campaign but he played all the Champions League knockout matches -- including in the 1-0 final defeat to Bayern Munich -- but was not offered a new deal after eight years at the Parc des Princes.
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The Brazil centre-back has agreed personal terms with Chelsea, although minor details are still left to sort out. He will speak to manager Frank Lampard on Monday once he is back in Paris from Lisbon.
The big attraction for Chelsea, who are also expected to sign Ben Chilwell from Leicester for a fee around £50 million and are closing in on the signing of Kai Havertz from Bayer Leverkusen on top of the deals already done for Timo Werner and Hakim Ziyech, is they won't have to pay a transfer fee for the former AC Milan defender.
Silva, 35, feels he can still play at the top European level for three or four years. He wanted a two-year deal and Chelsea were happy to offer it him.
PSG boss Thomas Tuchel wanted to keep him in Paris but sporting director Leonardo was against it.