The Costa Rica international established himself as the Spanish giants' first-choice goalkeeper after his arrival in 2014 from Levante and helped Madrid to three consecutive Champions League titles between 2016 and 2018.
"If God changes our plan it is because he has something better in mind," Navas said in a tweet. "I hope to begin a new path and make history with PSG."
Navas, 32, also won the La Liga, Supercopa de Espana, two UEFA Super Cups and three FIFA Club World Cup titles but has since fallen to second in the pecking order behind Thibaut Courtois.
Navas, who underwent his medical in Spain, should debut for the French champions at home to Strasbourg in Ligue 1 after the international break.
He will also immediately come up against his former employers Madrid in the opening match of PSG's Champions League Group A campaign at Parc des Princes.
Areola, 26, moves to the Bernabeu on a one-year loan deal with no option to purchase.