Mbappe recently stated that he planned to remain at PSG next season but hinted that he may not extend his deal at the Parc des Princes.
"I said that next year I will play for PSG. I still have a year left on my contract and I am going to honour my contract," Mbappe said at an awards ceremony last month after being named Ligue 1's player of the year.
The 24-year-old France striker signed a new deal a year ago to stay at PSG, turning down an approach from Real Madrid in the process.
But that contract will expire in 2024 unless he decides before July 31 this year to agree an extension for a third season, until 2025.
If he does not sign, the French champions therefore risk losing him for nothing in a year's time, meaning they could now opt to sell him in the current transfer window, which is open until September 1.
Mbappe is currently with the France squad preparing for Euro 2024 qualifying matches against Gibraltar in Portugal this Friday, and at home to Greece next Monday.
Mbappe's decision not to renew, announced in a letter to the club and first revealed by French sports daily L'Equipe, is a major blow to PSG.
It comes as they have been planning to make the forward the focal point of their team after allowing Messi to leave at the end of his own two-year contract.
According to sources with knowledge of the discussions, PSG were surprised that the letter was leaked and do not envisage letting Mbappe leave for nothing.
That means either renegotiating the contract or selling him in this transfer window.
"PSG play hardball" was the headline on the front page of L'Equipe on Tuesday.
Mbappe's entourage said they would make "no comment for the moment" when contacted by AFP.
He helped France to World Cup glory in 2018 and has established himself as arguably the leading forward on the global stage.
Mbappe also scored a hat-trick in France's 2022 World Cup final defeat to Messi's Argentina before helping PSG win a record 11th French title.
He was the top scorer in Ligue 1 last season with 29 goals and netted 41 in 43 games for his club in all competitions.
Yet it was an often frustrating season, as PSG went out of the Champions League in the last 16 to Bayern Munich and lost at the same stage in the French Cup.
Their regularly poor performances in the second half of the campaign persuaded PSG not to offer new terms to Messi, who fell out with the club after missing training to take an unauthorised trip to Saudi Arabia.
Last week Messi announced that he would move to MLS side Inter Miami.