PSG had not planned to recruit a goalkeeper this summer, preferring to remain with Navas, but the opportunity to sign Donnarumma on a free agent deal was considered too good to turn down.
However, Navas is expected to start the campaign as the first-choice goalkeeper.
French newspaper L'Equipe reported on Friday that Navas, who signed a new contract last month, was furious upon hearing that the club were set sign to Donnarumma this summer but has since been persuaded to fight for the No. 1 spot.
There is a small chance that Donnarumma will go back to the Serie A on loan, but Pochettino is preparing to keep two senior goalkeepers in his squad this season.
Donnarumma's arrival will mean PSG have nine goalkeepers under professional contract at the club. The list also includes Alphonse Areola, who is back from his loan spell at Fulham and Sergio Rico, who played backup to Navas last season. The French giants are trying to offload both goalkeepers.