The Brazil international moved to the French capital – along with Zlatan Ibrahimovic – from Serie A giants Milan in 2012.
Silva went on to make over 300 appearances and helped PSG win seven Ligue 1 titles, the Coupe de France five times and the Coupe de la Ligue on six occasions.
The 35-year-old's contract expired following PSG's 1-0 defeat to Bayern Munich in the Champions League final on Sunday and the club have now officially confirmed his exit.
The centre-back seems set for the Premier League as Chelsea are expected to sign the free agent in the coming days, according to reports.
PSG president Nasser Al-Khelaifi paid tribute to Silva, who he declared will go down as a legend.
"Thiago, thank you for these eight memorable years, full of lasting memories and for your involvement, your devotion and your aura," PSG's official Twitter account posted on Thursday.
"You will forever be one of the greatest players in history, and your legendary status here at PSG will last forever.
"I wish you and your family the best for your future challenges, you will always be part of the family and the history of the club, a big thank you, captain!"
Silva had also been linked with a move to Fiorentina but appears on course to be the latest signing in a busy transfer window for Chelsea.