The France international has now entered the final year of his contract at Stamford Bridge and is understood to have turned down the Blues' first extension offer.
Chelsea co-owner Todd Boehly is supposedly keen to keep Kante on the books, but the 31-year-old is holding out for a better offer than the two years that the West London club offered him.
The verbal proposal that was turned down by Kante was understood to have included the option for a third year, but the midfielder reportedly wants a guaranteed contract until 2025 with a potential 12-month extension until 2026.
With a breakthrough yet to be made in contract discussions with the Blues, Kante will be free to negotiate a pre-contract agreement with foreign clubs in January and is not expected to be short of suitors despite his injury record.
According to 90Min, PSG have now touched base with Kante's representatives to sound out a possible move in 2023, with Christophe Galtier keen to bring him back to his homeland.
The Ligue 1 champions underwent a significant midfield overhaul in the summer transfer window, bringing in no fewer than four engine room additions in Vitinha, Carlos Soler, Renato Sanches and Fabian Ruiz.
Vitinha and Marco Verratti have formed Galtier's first-choice pairing so far, while Georginio Wijnaldum, Ander Herrera, Julian Draxler and Leandro Paredes left on loan alongside the permanent exits of Rafinha, Xavi Simons and Idrissa Gueye.
Kante plied his trade at Caen, US Boulogne and Suresnes back in France before making the move to Leicester City in 2015, but a separate report has poured cold water on speculation surrounding a move to PSG.
As quoted by Le 10 Sport, journalist Daniel Riolo has claimed that Kante will "never" go back to playing in Ligue 1 due to "problems" with a particular group of people, saying: "Over the past five years, almost every summer, the transfer of Kanté to PSG has been mentioned by the media.
"When I inquired about the feasibility of the transfer, each time "one" told me: 'Kante will never return to France, he has too many problems with people from 92' Difficult to know more."
Even if a move to PSG falls through, Kante is still expected to be the subject of interest from Premier League clubs, with Arsenal rumoured to be keeping tabs on his situation.
The World Cup winner has tallied 13 goals and 15 assists from 262 appearances in a Chelsea shirt, but he has struggled with persistent injury problems over the past couple of years.
As well as Kante, Jorginho is currently due to leave Chelsea on a free transfer in 2023 and has supposedly offered himself to Barcelona, but the Catalan club are not interested in signing him at this moment in time.
Graham Potter also has Conor Gallagher, Ruben Loftus-Cheek, Mateo Kovacic and Carney Chukwuemeka as his midfield options, but the Blues are alleged to have turned down the chance to sign Aurelien Tchouameni before his switch from Monaco to Real Madrid.