The LaLiga side's proposal -- worth €170m plus €10m in add-ons -- was submitted on Thursday, with optimism so strong at Madrid that some sources told ESPN that an agreement will be reached within a matter of hours.
PSG executives have not rejected this second offer, which was raised from an original €160m bid from Tuesday, with wishes that Mbappe would remain and form a potent attack alongside Neymar and newly arrived Lionel Messi.
PSG hoped that World Cup winner Mbappe, who is out of contract in June 2022, would extend his deal at club. However, that now appears to not be the case and PSG will look to cash in on the player's transfer instead of risk losing him as a free agent.
Earlier on Thursday, PSG president Nasser Al-Khelaifi reiterated their stance on Mbappe, saying: "I think that our position is very clear, as [sporting director] Leonardo said yesterday. For Kylian, our position has not changed. Everyone knows it, Kylian knows it. Our position is very clear and honest."
Mbappe has been a long-term, first-choice target for Real Madrid as president Florentino Perez -- who is personally overseeing the negotiations -- looks to build a new team to shine at the redeveloped Bernabeu stadium.
The club has known for months that the 22-year-old was keen to join, and ESPN reported in September that careful preparations were being made to make a move for Mbappe this summer.
However, at the same time, Madrid have been desperate not to act in a way that could anger PSG and threaten a possible deal, as well as the relationship between the two clubs.
Madrid have brought in around €80m from the sales of Raphael Varane to Manchester United and Martin Odegaard to Arsenal this summer and have not paid a transfer fee for a first-team player since 2019, choosing instead to save up for when the moment came to bid for Mbappe.