Newspaper L’Equipe and radio station FranceInfo claimed the Frenchman had agreed to replace Mauricio Pochettino at Parc de Princes.
A source close to the club told AFP later that “numerous discussions are underway”, despite the reports.
Argentinian Pochettino, who was appointed in January last year and led the club to the Ligue 1 title in the season just ended, has another 12 months remaining on his contract.
French newspaper have reported however that Pochettino and the club to come to an agreement that would see the Argentine leave the French capital.
Galtier’s current employers were unavailable for comment.
If he does make the move to Paris, Galtier would reunite with Louis Campos, PSG’s recently appointed football advisor.
Earlier this month, French press linked former Real Madrid boss Zinedine Zidane with the position, with PSG’s Qatari owners desperate to win the Champions League.
France World Cup winner Kylian Mbappe signed a new deal with the Parisians last month, choosing against joining the Spanish giants.
The new French top-flight season begins on the weekend of August 6-7.
A week before that, champions PSG will play French Cup winners Nantes in the Trophee des Champions, the traditional curtain-raiser to the French season, in Tel-Aviv, Israel.