The Ligue 1-winning Frenchman has arrived at the Parc des Princes to replace Mauricio Pochettino, whose departure from the capital club was made official earlier this month.
Galtier has reunited with former Lille sporting director Luis Campos, who worked alongside the 55-year-old during their 2020-21 title-winning season with Les Dogues.
PSG have already won the race to sign highly-rated midfielder Vitinha from Porto, as well as making Nuno Mendes's move from Sporting Lisbon permanent.
Angel Di Maria, Xavi Simons, Alphonse Areola and Marcin Bulka represent some of the French champions' summer outgoings, while Colin Dagba has been loaned out to Strasbourg.
According to Marca, Galtier could be willing to sanction the exits of 11 players in the coming weeks, with Neymar representing the most high-profile possible outgoing.
The 30-year-old struggled with injuries during the 2021-22 season and was also booed by his own supporters in the wake of PSG's last-16 Champions League exit to Real Madrid.
However, Galtier insisted at his first press conference that he wanted to keep working with Neymar, telling reporters: "Which coach would not want to have him in their team? I have an idea of exactly what I expect from him. I hope that he stays with us."
Newcastle United are thought to be the only team that could afford a transfer fee and Neymar's extortionate wages, but the Brazil international has no desire to move to St James' Park.
The report adds that a number of midfielders could also depart the Parc des Princes this summer, namely Leandro Paredes, Georginio Wijnaldum and Ander Herrera.
Wijnaldum largely failed to impress during his first season at the club after leaving Liverpool on a free transfer, and the Dutchman has been widely linked with a return to the Premier League.
Exits for fellow midfielders Rafinha and Eric Junior Dino Ebimbe could also be sanctioned, with both players due to become free agents at the end of next season.
Layvin Kurzawa and Abdou Diallo are also being linked with moves away, as Mendes resumes his battle with Juan Bernat for the left-back spot in the 2022-23 campaign.
Julian Draxler and Mauro Icardi have also failed to become regular fixtures of PSG's star-studded forward line, although the latter has rubbished talk of a move away from the Parc des Princes.
Les Parisens have been linked with a shock move for Cristiano Ronaldo in recent days, with the Manchester United attacker supposedly pushing for an exit from Old Trafford due to concerns over the club's ability to challenge for silverware.