The 55-year-old spent just one season in the Aiglons hotseat after leading Lille to the 2020-21 Ligue 1 title alongside newly-appointed PSG advisor Luis Campos.
Galtier helped Nice return to European football with a fifth-placed finish in the 2021-22 top flight, which was enough for a spot in the Europa Conference League playoff round.
However, Les Aiglons fell short in their bid to qualify for the Champions League, as Galtier's side won just six of their last 13 games in the Ligue 1 campaign.
The Frenchman's relationship with the Nice board had allegedly soured over the past few weeks, and director Julien Fournier had opened the door for PSG to make their move.
""If Paris-Saint-Germain want Christophe Galtier, they mustn't wait too long, otherwise it will be more complicated," Fournier told L'Equipe.
"He's under contract, that's the reality. So he's more likely to be at Nice than not to be. It's very basic but it's very true. After that, if Paris-Saint-Germain comes in for him... it's a decision that I obviously won't take alone.
"When this type of club comes to get one of your players or your coach, it's hard to say 'no'... Each passing day makes PSG's task more difficult, because we have to move forward with next season. If they want him, they must not delay."
Galtier is now one step closer to his PSG appointment, with Nice confirming the 55-year-old's exit from the club on Monday afternoon as Lucien Favre takes his place.
PSG are understood to have agreed a €10m (£8.6m) to bring Galtier to the club, with the reigning champions buying him out of his contract, which was due to expire in the summer of 2024.
The former Saint-Etienne manager will arrive at the club to replace Mauricio Pochettino, who will leave the Parc des Princes after just one-and-a-half seasons in charge.
Pochettino won the Ligue 1 title, the Coupe de France and the Trophee des Champions in the French capital, and his sacking should be made official within the coming days.
Galtier won 24, drew seven and lost 12 of his 43 games in charge of Nice across all tournaments and also led Les Aiglons to the final of the Coupe de France, where they lost to Nantes.
Meanwhile, 64-year-old Favre has been out of management since leaving Borussia Dortmund in December 2020 and previously spent two years in charge of Nice between 2016 and 2018, during which he led Les Aiglons to a third-placed finish in his first season in charge.