A source told AFP that 31-year-old Neymar "no longer figures in the club or manager's plans" with the Brazilian immediately linked to Al-Hilal in the Gulf kingdom.
Another source in Saudi Arabia said that "direct negotiations" between Neymar's agents and the Public Investment Fund (PIF) were ongoing.
"If we reach an agreement with him in the coming hours, we would start negotiating with PSG immediately. He would play for Al-Hilal if a deal is sealed," said the source.
Neymar joined PSG for a record €220 million (£190 million) from Barcelona in 2017.
Last month Al-Hilal made a €300 million bid for Mbappe, though the striker reportedly refused to meet with officials from the team.
Neymar underwent surgery on his right ankle in early March, only returning to join PSG on their pre-season tour of Asia.
His time at PSG has been blighted by a catalogue of injuries.
Although he helped the club to the 2020 Champions League final he has often been sidelined for key games.
If Neymar does end up moving to Saudi Arabia he will join a growing list of top players lured to the oil-rich kingdom since Cristiano Ronaldo moved to Al-Nassr in January.
It has been a rocky build-up to the new Ligue 1 season for champions PSG with both Mbappe and Neymar left out of new coach Luis Enrique's team for Saturday's underwhelming opening goalless draw against Lorient.
After being sidelined for several weeks amid an ongoing contract dispute Mbappe watched from the Parc des Princes stands.
But the club's prize asset is now back in the fold, for the time being at least, said PSG on Sunday.
"After very constructive and positive discussions between Paris Saint-Germain and Kylian Mbappe before the PSG-Lorient match, the player was reinstated in first team training this morning," PSG said in a statement.
Mbappe stand-off
After a summer of flux at the Parc des Princes filled with new arrivals, doubts over Neymar's future and Lionel Messi's departure, it is the Mbappe stand-off that dominated the headlines.
The France captain has refused to sign an extension to his PSG deal, meaning he can leave for free next year, with Real Madrid long seen as his preferred destination.
Qatari-owned PSG want to sell him now and bring in a significant transfer fee for a player who cost 180 million euros from Monaco in 2017.
"The position is very clear. If Kylian wants to stay, he must sign a new contract. We can't let the best player in the world today leave for free. It's impossible," PSG president Nasser al-Khelaifi said last month.
Mbappe had not been allowed to train with the first team and played only one of PSG's five warm-up games and was not invited to the pre-season tour in Japan and South Korea.
He sat alongside fellow 2018 World Cup winner Ousmane Dembele, who completed his transfer from Barcelona earlier Saturday, as Luis Enrique handed debuts to six players against Lorient.
