Navas, a three-time Champions League winner with Real Madrid, fell out of favour in the French capital with Gianluigi Donnarumma the preferred goalkeeper for Christophe Galtier's side.
The 110-cap Costa Rica international will add a wealth of experience as Cooper's side attempts to stave off Premier League relegation.
Navas, who won two Ligue 1 titles with PSG after leaving Madrid in 2019, could be in line for his Forest debut in a crucial relegation clash at home to Leeds United on Monday (AEDT).
The arrival of Navas, who Forest has the option to sign permanently at the end of the season, was announced shortly after confirmation of Jonjo Shelvey's signing from fellow Premier League side Newcastle United.
Shelvey has penned a two-and-a-half-year contract with Forest, with the fee undisclosed but expected to be insignificant as the midfielder's Newcastle deal was set to expire in June.
Former Liverpool man Shelvey suggested Cooper was a big draw after discussing the move with Newcastle manager Eddie Howe.
"It's a fantastic football club that has so much history and I'm really looking forward to getting started," Shelvey said. "My main goal straight away is to fight for my place in the team and stay there.
"It's not going to be easy because there's a lot of competition for places, but it's a challenge I'm really looking forward to and the competitiveness can only help the team going forward.
"I spoke to Eddie Howe and he had such good words to say about Steve Cooper. I previously knew of him from my time at Liverpool, so I knew what a fantastic coach he was, but meeting him for the first time has been a breath of fresh air.
"He's told me what he expects and the high standards he sets, so I can hope I can come and help the team push to stay in the Premier League.
"I've been in this league a long time, I know what it takes to stay up and what it takes to get a good team to gel, so I'm looking forward to getting started and working with the players that are here."