France July 25 2023: The 24-year-old France captain, who has a year left on his contract, has refused to sign an extension at the French champions and was not selected for their pre-season tour to Japan.
PSG want to sell Mbappe now rather than see him leave for free next summer.
Neymar's £200m move from Barcelona to PSG in 2017 is the current record.
PSG activated Brazil forward Neymar's release clause to make him the most expensive player in the world.
Should he move, Mbappe will have commanded two of the three highest fees paid for a player after joining PSG from Monaco for £166m in 2017.
Chelsea, Manchester United, Tottenham, Inter Milan and Barcelona have all expressed an interest in Mbappe, who has previously said he intends to leave PSG for free at the end of next season and is believed to prefer a move to Real Madrid.
PSG are open to the idea of Mbappe leaving on loan this summer to see out the final year of his contract elsewhere.
This would allow him to join Real Madrid for nothing in 12 months' time, keep playing this season and generate a loan fee for PSG.
Chairman Nasser Al-Khelaifi has made clear his determination to take a tough stance and there have been suggestions Mbappe would not be selected for an entire year if he refused to break the present impasse. That eventuality would clearly have a negative impact on his hopes of playing for France at Euro 2024.
Al-Hilal are one of the clubs owned by Saudi Arabia's Public Investment Fund. They have already signed Ruben Neves from Wolves and Kalidou Koulibaly from Chelsea this summer.
Timeline of events
- May 2022: Mbappe signs lucrative two-year deal with the option of a further year to stay at PSG, turning down a move to Real Madrid.
- December 2022: Scores hat-trick but France beaten on penalties in World Cup final by Argentina.
- March 2023: Named captain of France, replacing the retired Hugo Lloris.
- April 2023: Wins sixth Ligue 1 title of career and fifth with PSG.
- 13 June 2023: Mbappe tweets he wants to stay at PSG for now but will not extend his contract beyond 2024.
- 5 July 2023: Told by PSG president Nasser Al-Khelaifi he must sign a new contract if he wants to stay.
- 9 July 2023: Calls PSG a "divisive" club in comments made to France Football magazine.
- 21 July 2023: Left out of PSG squad for pre-season tour of Japan and South Korea.
- 22 July 2023: French footballers' union says it could take legal action after accusing PSG of "moral harassment" for leaving Mbappe out of pre-season tour squad.
- 24 July 2023: Saudi Arabian side Al-Hilal make world record £259m bid for Mbappe.