But there is said to be a US$255 million in his contract and the French club, who are owned by an investment fund headed by the son of the Emir of Qatar, are thought to have agreed to pay the fee, which would almost double the world record for a transfer fee.
The great @NeymarJr attempts a terrifying roof-to-roof shot high over Hollywood Blvd! pic.twitter.com/VYuXVkptg5
— Jimmy Kimmel (@jimmykimmel) 8 June 2017
Barcelona, for their part, insist the player is staying at the Nou Camp stadium.
Football journalists around the world are divided as to whether Neymar will actually join PSG but if he does it will mark a staggering jump in transfer fees.
Only a week ago Manchester United agreed to pay US$97 million up front for Belgian striker Romelu Lukaku in a deal which could eventually cost them up to £100 million (US$129 million).
Last summer, United spent 105 million euros (US$136 million) on French midfielder Paul Pogba from Juventus.
United are one of the richest clubs in the world, but they have not won the Premier League since 2013.
Barcelona failed to win the Spanish league last season, coming second to arch-rivals Real Madrid and their club president, Josep Maria Bartomeu, has insisted Neymar will part of their team as they try to win back the title next season.
The Brazilian, who scored 13 goals last season, only signed a five-year contract in October.
LA turning up! 📶🔊🔥⚽️ thank you, East Los, for an incredible Night. #NeymarLive #NJRmixtape @Nikela @nikefootball pic.twitter.com/QMVJL8h7VR
— Neymar Jr (@neymarjr) 8 June 2017
But the contract reportedly included the huge buyout clause, which was no doubt inserted on the assumption that no club would be willing to stump up such an enormous sum of money.
Neymar joined the Catalan giants in 2013.
He famously fractured a vertebrae in his back during the 2014 World Cup in Brazil and was unable to play for the host nation as they crashed out to Germany, losing 7-1 in the most humiliating night in Brazilian footballing history.
The question now is whether Neymar wants to force Barcelona's hand and drive through a move to the French capital.
Last year he was facing a two year jail sentence over alleged corruption involving his transfer from Brazil. But that issue seems to have gone away and he seems happy in Catalonia.
"The last one was my best season in Barcelona. I am very adapted to the city, the club and I am happy here. That is reflected on the performance of an athlete," he told Goal.com.
"We played great games and lived incredible moments. Now, we need to work for the 2017-18 season so it will be even more positive individually and as a group," he added.
Neymar has 30 million followers on Twitter but has said nothing about the overtures made by Paris Saint-Germain.
📸👍 @marquinhos_m5 @psgclubmontreal 👊🔴🔵 #RomaPSG #ParisLovesUS 🇺🇸 pic.twitter.com/ucDoNnoDRF
— PSG Officiel (@PSG_inside) 19 July 2017
The French club was taken over by the Qatari consortium in 2011, and they won four successive French league titles, but lost out to Monaco last year.
Paris Saint-Germain lost an incredible Champions' League match to Barcelona in March.
PSG led 4-0 from the first leg in the French capital, but Barcelona won the second leg 6-1, with three goals in the last five minutes, including two from Neymar.
Barcelona were themselves later knocked out by Juventus.
