Silva dropped into the free agency pool after lining up for PSG in the Champions League final on Sunday.
Thomas Tuchel's side lost 1-0 to Bayern Munich in the European showpiece in Lisbon, but still finished their latest campaign with four major trophies, taking the Brazilian defender's overall haul of silverware at the club to 23.
The 35-year-old was not offered the chance to extend his stay at Parc des Princes, and is currently on the lookout for a new challenge after spending the last eight years of his career in the French capital.
Chelsea are believed to leading the race for Silva's signature, with the former PSG centre-back reportedly set to undergo a medical at Stamford Bridge later this week.
Silva is expected to commit his future to the Blues through to 2023, and a fellow countryman is certain that he will add a new dimension to Lampard's current squad while also serving as a role model for the club's younger players.
Brazil legend Rivaldo, who picked up a World Cup winners' medal in 2002, told Betfair: "At 35, Thiago Silva will sign for two seasons with Chelsea and I'm confident that he will do a great job there, as he is a great centre back with excellent positioning and vast experience of world football.
"I think he still comes with plenty to offer. I'd back him to produce a good football in the Premier League and should be a commanding voice in Frank Lampard's otherwise young squad.
"Nowadays, many players extend their careers until their late-thirties and Silva is a good example of that, as I was in my time.
'If you keep yourself in good physical condition you can easily play at the highest level until 36, 37 years old."
“It makes sense for me," Burley told ESPN earlier this week. "It’s not ideal signing a [soon to be] 36-year-old, but it’s a fit 36-year-old who’s got good character, good experience and can still play.”