The 23-year-old forward is set to leave Paris Saint-Germain on a free transfer this summer with his contract set to expire in June.
And it looks like his next destination will be the Bernabeu, however, The Athletic reports Mbappe has not yet signed any paperwork with Real Madrid.
The report adds the Ligue 1 champions are still trying to convince him to change his mind.
On his future, Mbappe said at the weekend: "I can't say about my future but you'll know very soon… it's almost decided.
"This isn't the right moment but yes, yes… my decision is almost done."
Los Blancos have been interested in the World Cup winner for some time and tried to sign him last summer, however, PSG rebuffed their big-money bids.
But now he's set to join the LaLiga giants for free. Real Madrid are also set to land Chelsea defender Antonio Rudiger on a Bosman, while Manchester United midfielder Paul Pogba has been heavily linked with a move this summer too.
Mbappe is regarded as one of the finest forwards in the world. He's scored 116 goals in 141 appearances for PSG.
He joined PSG from Monaco, who he helped win the Ligue 1 title, in a deal understood to be worth £165million. Mbappe boasts five French league titles to his name.
The player has enjoyed huge success on the international stage having won the World Cup and Nations League with his native France in 2018 and 2021 respectively.
He's also collected a number of individual accolades having been named Ligue 1 Player of the Year three times, being selected in the Champions League Squad of the Season three times, while also being named the best young player at the 2018 World Cup.
His prospective new club have enjoyed a successful season having won their 35th LaLiga.
They also have a chance at becoming European champions for a 14th time as they face Liverpool in the Champions League final on May 28.