Widespread reports on Wednesday (AEST) revealed a €160million offer from Madrid for Mbappe, considered to be its top target.
A bid was confirmed by PSG sporting director Leonardo on Thursday – although he would not detail the exact amount – and has been rejected by the Ligue 1 giant.
Leonardo discussed the potential transfer at length with the media, outlining his anger at Madrid's approach for a player who has scored 133 goals and provided 58 assists across 174 games over four years at PSG.
The club would clearly like to keep Mbappe, whose 191 goal involvements since his debut rank third among players in Europe's top five leagues in that time, behind new team-mate Lionel Messi (236) and Robert Lewandowski (213).
But the World Cup winner, whose contract expires next year, has his sights set on the Santiago Bernabeu.
Leonardo said: "This seems clear, doesn't it? If Real Madrid make an offer a week before the end of the [transfer] window, it is because the player wants to leave.
"If Kylian wants to speak out and say he doesn't want to leave, then he won't leave, because that's our wish.
"Real have talked with the family, the entourage, otherwise they would not have done so [made the bid]."
This late-window saga comes as a real blow to PSG, who had enjoyed surely the best close-season of all leading European clubs.
Messi followed Gianluigi Donnarumma, Sergio Ramos, Achraf Hakimi and Georginio Wijnaldum in heading to Paris. Mbappe has not yet featured alongside Messi, who could make his PSG bow this week.
"We cannot be happy with this situation, because we did something this summer, we have created a dream, we will not let anyone destroy it or position themselves above the club," Leonardo added.
"Kylian is at the centre of the project but not above it. We created a positive atmosphere, a desire, an emotion. That was our plan."
Neymar, like Messi, is yet to feature this term, with Mbappe holding the fort in their absence, weighing in with a team-high three goal involvements in three games.
Cristiano Ronaldo, said to be keen to leave Juventus, would appear to be one potential replacement if Mbappe did leave, but Leonardo has not given it any thought.
"We never thought of a plan without Kylian," he said.