The teenager, tipped to be a future Ballon d’Or winner, will move on loan for a season – but there is an obligation to buy next summer that will see him move for £166million.
The 18-year-old was left on the bench as Monaco smashed Marseille 6-1 on Sunday evening.
The France international has only played one game so far this season because of a knee injury he suffered in the opening game.
Mbappe has been linked with a move away from the Stade Louis II all summer.
Real Madrid, Manchester United, Manchester City and Barcelona were all linked.
But it appears that PSG – who broke the world record to sign Neymar earlier this month – have beaten the other clubs to sign Mbappe.
And they seem to be getting around Financial Fair Play at the same time with the loan deal.
Monaco already seem to have lined up a replacement.
The unbeaten Ligue 1 club are looking at Manchester City flop Stevan Jovetic.
The Montenegrin, 27, has struggled since moving to Inter Milan and spent last season out on loan at Sevilla.