Real Madrid made a 137m offer to Paris St-Germain forward Kylian Maappe.
22-year-old French World Cup winner joined PSG in 2017, in a transfer of 165.7m. His contract will expire in June 2022.
Mbappe scored 133 goals in 174 matches for PSG, winning three Ligue 1 titles as well as three French Cups.
Real Madrid has a debt of approximately 1bn and have signed David Alaba, a defender, on a free transfer by Bayern Munich.
High-earning Sergio Ramos, Raphael Varane, and Manchester United's Raphael Varane, have left. Real, however, feels sufficiently secure about their financial situation to approach one of the top-rated players in the game.
It is unknown how the fee will be structured.
PSG signed Argentina forward Lionel Messi this month, after he had left Barcelona.
Real could be successful and trigger several deals before the deadline.
PSG is believed to be interested in Cristiano Ronaldo, whose future at Juventus is uncertain. This raises the possibility of a link up with Messi, his rival for the title of best player in the world for well over a decade.
Manchester City were among the clubs that Ronaldo was offered, but they aren't interested in a deal.
Last week, both Juventus boss Massimiliano Allgri and Ronaldo denied that the forward wanted to go.
"The deal is confusing people"
Robin Bairner, a French journalist for football, on BBC Radio 5 Live
The French are still baffled by the way this deal works. It is both puzzling to see how PSG could reject such a large bid for a player out of contract in less that 12 months and how Real Madrid can afford so much money.
We are not sure how the deal will be structured. There has been no indication from Real, at least from France, of how Real propose to pay 160m euros. It is unclear if it is 80m upfront and 80m in bonus payments.
I don't believe anyone expects them to pay 160m upfront and Mbappe's weekly wages, which exceed half a million euro per week.
PSG considers prestige more important than money for their Qatari owners. Mbappe is their most valuable asset. He's a man they don't want lose, and it's a fight PSG is not used to losing. They have been able to buy anyone they want in the past and Mbappe is not resisting. In the 10 years that they have been in charge, it is somewhat alien to them.
According to me, PSG did not want Cristiano Ronaldo signed.
Although they were determined to keep Mbappe, their stance on Mbappe seems to have changed a little in the last few days, at most behind closed doors, if not publicly.
They are considering other options; Everton's Richarlison is one of them. They can't sign Ronaldo over Mbappe because of his wages, but PSG is a club where it doesn’t seem important at times. It might be possible to sign Ronaldo for a few years if they receive 160m for Mbappe.