The French club realise the importance of the 23-year-old to their project, as perhaps the best player in the world for the next decade, and are willing to offer the sort of contract not yet seen in the game.
It is believed that PSG would pay him a basic wage far in excess of £500,000, and closer to £1m a week. Real Madrid, who Mbappe is most interested in joining, would not be able to get close to such an offer.
Some of Mbappe's family and entourage had scheduled meetings with PSG on Wednesday, the day after his match-winning display against the Spanish club. In that meeting, the staleness of the Madrid team compared to PSG was remarked upon.
Bernabeu president Florentino Perez - who was "furious" with the nature of his teams performance and considering replacing Caro Ancelotti with Mauricio Pochettino - is currently banking on Mbappe re-igniting the whole club.
Nothing is as yet signed with Madrid, though, and the player and his camp are at least receptive to considering PSG's offer. His contract runs out this summer, and he could already sign a deal with another club now, but has so far left his options open.
Mbappe has been captivated by the idea of signing for a club like Madrid, given Cristiano Ronaldo is one of his heroes. The Bernabeu hierarchy are hoping that the player's emotional desire to join such a "legacy, but Mbappe is also aware of the necessity of absolutely maximising his talent in terms of trophies. Spanish football may not be as conducive to that given its current economic crisis, which pushed Madrid to drive the Super League project.
At the same time, such a huge contract offer - by far the biggest football has ever seen - is a key part of the Qatari project at PSG, as they intend to put themselves in a position where only a handful of clubs can match them.
The situation may nevertheless influence other moves, and potentially mitigate the offer for Paul Pogba, who can similarly leave Manchester United on a free this summer. PSG were willing to offer Pogba up to €600,000 a week, but that was in part because they want a French - and preferably Parisian - star at the centre of their team.
That would not be so necessary if Mbappe stayed at the Parc des Princes. Some figures close to the situation remarked on the timing of stories on Wednesday night, where it was leaked that Pogba would be interested in proposals from other English clubs.