Mbappe has established himself as one of the best players in the world since joining PSG from French rivals Monaco in 2018.
The forward initially signed for PSG on loan but made the move permanent two years ago and has now racked up 90 goals in 124 appearances for the club.
Mbappe, 21, has already enjoyed substantial individual and collective success, winning three Ligue 1 titles and of course the World Cup with France.
PSG do not want to lose the superstar and, according to The Times, have made several offers in an attempt to persuade Mbappe to sign a new deal on improved terms.
But it is claimed that Mbappe has made it clear that he intends to stay at PSG for one more season before leaving next summer, possibly to join the Premier League.
The French World Cup-winner has been vocal in his admiration for defending Premier League champions Liverpool but The Times say he is also interested in a move to Manchester United.
Ole Gunnar Solskjaer has identified Jadon Sancho as his top target this summer but Mbappe could emerge as an alternative target in 12 months’ time if Manchester United fail to force through a deal before the October 5 deadline.
Discussing Jurgen Klopp’s Liverpool earlier in the year, Mbappe said: ‘What Liverpool do in this moment is amazing.
‘They’re like a machine, they’ve found a rhythm and are like “we play again, we play again”. When you watch you think everything’s easy, but that’s not easy.
Liverpool already boast one of the best front-threes in world football with Roberto Firmino, Mohamed Salah and Sadio Mane thriving over the past few seasons.
But all three players are approaching the age of 30 and Liverpool may consider tweaking their attack next summer, particularly if they fail to defend their title this season.
Liverpool kicked off their 2020-21 title defence with a victory over newly-promoted Leeds United, but it was far from straightforward, with the Reds needing to score four goals to earn the home win.
PSG, meanwhile, suffered a shock defeat to Lens in their season opener on Thursday.
Mbappe was one of a number of PSG stars to miss the match after testing positive for coronavirus.