Mbappe caused a stir when he sent a letter to the club stating that he had no intention to extend his PSG contract, which expires in 2024 if he does not activate the clause that will extend his deal by a further 12 months.
PSG find themselves in a fix as they risk losing Mbappe for free in June 2024. He will also be free to sign a pre-contract agreement with a new club from January should he decide to see out the remaining year of his deal at PSG.
"I didn't ask either to be sold or to go to Real Madrid," Mbappe told Italian publication Gazzetta dello Sport.
"I just confirmed that I don't want to activate the extra year foreseen in the contract. We have never talked about a renewal with PSG, but I'm happy to stay here next season."
However, if Mbappe stays until the end of the 2023/24 season, the French club will be unable to recoup any of the 180 million euros they spent in 2017 to sign the forward from AS Monaco.
Real Madrid have courted Mbappe in the past but failed to land the forward, who signed a contract extension with PSG in May last year.
However, the Spanish club are rebuilding their team after losing several attacking players, including Ballon d'Or winner Karim Benzema, and could turn their eye to Mbappe who has finished top scorer in Ligue 1 for the last five seasons.
PSG will also want to avoid a complete rebuild of their forward line after losing Lionel Messi on a free transfer when the Argentine World Cup winner opted not to extend his contract at the end of the season.
Messi had been booed and jeered by PSG fans towards the end of the season after their Champions League last-16 exit and Mbappe said the seven-times Ballon d'Or winner did not get the respect he deserved in France.
"We're talking about potentially the best player in the history of football. It's never good news when someone like Messi leaves," Mbappe said.
"Personally, I don't quite understand why so many people were so relieved that he was gone... It's a shame, but that's how it went. We will have to do what we can to replace him."