Those at Stamford Bridge were reluctant to match that bid, leaving the path clear for the Argentina international to head for Parc des Princes.
Both Chelsea and PSG had identified Paredes as a top target as respective bosses Maurizio Sarri and Thomas Tuchel require midfield reinforcements.
Paredes has now arrived to bolster the ranks at PSG, with the South American telling the club’s official website: “I’m very proud to be joining Paris Saint-Germain.
“After my previous experiences in Italy and Russia, I now have this fantastic opportunity, not only to discover a new league, but above all one to wear one the most prestigious club jerseys in the world.
“I’d like to thank the chairman and the coach for the confidence they have shown in me by signing me.
“We all know the extraordinary sporting dimension of Paris Saint-Germain, its players and staff. I share the same high ambitions and I will do everything I can to give my club what it expects of me as a midfielder.
“I also look forward to the passionate atmosphere of the Parc des Princes, as I grew up with football fever in Argentina.”
PSG chairman Nasser Al-Khelaifi added: “We are very happy to welcome Leandro Paredes into the Paris Saint-Germain family today.
“Leandro brings with him the traditional strengths of Argentinian football - a fighting spirit and excellent technical skills.
“He will bring a lot to our midfield and I am sure he will fit quickly into our coach's well-organised and ambitious team.
“Leandro's arrival will keep the momentum, both on the pitch and among our supporters, ahead of the major events coming up. Exciting times lie ahead.”