The 23-year-old is running down the final few weeks of his contract with the French champions and is due to become a free agent at the end of the 2021-22 season.
With Manchester City winning the race to sign Erling Braut Haaland, Mbappe is now widely expected to sign for Real Madrid on a free transfer ahead of the 2022-23 campaign.
There has been more optimism over a new PSG deal for Mbappe in the past few weeks, although the player's mother has come out denying the existence of an agreement with the French club over a new contract.
Mbappe was named the Ligue 1 Player of the Year for 2021-22 over the weekend, and the France international revealed that he had "almost" made a decision on his future at the ceremony.
"I made the mistake three years ago of taking over the ceremony. I want to participate in the ceremony, no look for glory. We will know really soon, it's almost done. My decision is made, yes, almost," Mbappe told the press.
"It's always a huge pleasure being here. It's an honour, which is why I insisted on coming. Winning for a third consecutive time is incredible. I want to thank PSG, my teammates, my coach, and all those people working behind the scenes.
"We are proud of our tenth title. This year, more than ever before, I felt recognition and admiration. Thank you all. I will continue my story. I'm never satisfied."
Now, Marca claims that Mbappe is on the brink of signing a five-year contract at Real Madrid having recently been spotted in the Spanish capital.
The report adds that a "full agreement" over Mbappe's transfer was struck last week, but Real Madrid will not officially announce his arrival before the player has confirmed his departure from PSG himself.
Real Madrid's general director Jose Angel Sanchez is said to have taken charge of the process to bring Mbappe to the Santiago Bernabeu, with president Florentino Perez not speaking with the attacker directly at all.
Los Blancos are set to land Mbappe on a free transfer after PSG stalled on a £189m bid last summer, and the former Monaco starlet will supposedly earn less at the Bernabeu compared to the terms in Les Parisiens' renewal offer.
Mbappe - who has won four Ligue 1 titles and three Coupe de France crowns with PSG - is therefore set to make his final appearance for the club in Saturday's Ligue 1 clash with Metz at the Parc des Princes.
The World Cup winner has featured 216 times for the French outfit since his initial loan move from Monaco in 2017, scoring 168 goals and setting up 87 more.
Mbappe has already registered 36 goals and 26 assists in 45 appearances for PSG during the 2021-22 campaign, including a league-high 25 goals and 19 assists in Ligue 1.