The 18-year-old Brazilian was the subject of a summer approach from the Blues, who regard him among the best young players in the world in his position.
But Chelsea now face missing out on Moscardo after PSG made their move. PSG director Luis Campos is due in Brazil this weekend in an attempt to tie up a potential £21million deal.
PSG's approach comes amid late trepidation from Moscardo about Chelsea's current midfield options.
Chelsea have a glut of players in the position, with Enzo Fernandez, Moises Caicedo, Conor Gallagher, Romeo Lavia, Lesley Ugochukwu, Carney Chukwuemeka and Andrey Santos all aged 23 or younger.
They also cannot offer a potential loan pathway this season, having used up all seven international loan spots, including three for Kepa Arrizabalaga, Romelu Lukaku and Hakim Ziyech.
PSG are expected to advance in negotiations over the weekend, with Chelsea switching focus to a potential striker signing in January.
Mauricio Pochettino continues to assess whether he has enough talent in Christopher Nkunku, Nicolas Jackson and Armando Broja, or whether another forward option would boost Chelsea.
Moscardo was asked about Chelsea's approach earlier this month.
He said: "There was a proposal from Chelsea months ago, but I talk a lot with my agent and my parents and they try to give me as little information as possible.
"They say there is not long until the end of the season; they tell me to dedicate myself and do well and that there will be good things at the end of the season."
Arsenal and Barcelona have also been linked with Moscardo, who is keen to finish the season in Brazil before moving in January.