Madrid-based newspaper Marca claim that the Spanish and European champions have struck a deal that would easily eclipse the world-record sum of €105m that Manchester United paid to Juventus to sign Paul Pogba previous year.
According to the report, Mbappe will cost the Spanish giants €160 million, with another €20 million potentially being owed through performance-based clauses.
Having recently sold Alvaro Morata and Danilo, and having loaned out James Rodriguez, Real have freed up the necessary funds to move for the teenage sensation.
And Marca say Real have reinvested that figure and some by landing Mbappe on a six-year contract worth around £120,000-a-week.
Victor Moses relaxes in pool with Chelsea teammates
The German champions suffered a heavy 4-0 loss to AC Milan on Saturday and suffered defeat on penalties to Arsenal in Beijing last Thursday.
And so, it is likely that Benzema could lose his place in Zinedine Zidane's first-team setup, meaning he could be set to call time on his stint with Real Madrid this summer.
Real Madrid have been confident about signing the youngster all summer due to the player's reported dream of playing for the European champions next season.
Real Madrid met with Monaco last week and the Ligue 1 champions demanded 190 million euros for Mbappe.
Manchester City attempted to hijack the deal for Mbappe last week and this was behind Monaco's statement last week regarding tapping up. Moreover, Pep Guardiola's side will probably be trumped by the money PSG are offering for Sanchez.
Kylian Mbappe of Monaco celebrates his goal during the French Ligue 1 match between AS Monaco and AS Saint-Etienne (ASSE) at Stade Louis II on May 17, 2017 in Monaco, Monaco.