The former Monaco starlet has been relentlessly pursued by Los Blancos over the past couple of years and was the subject of two late bids from Carlo Ancelotti's men in the summer of 2021.
PSG were allegedly offered as much as €200m (£173.5m) to let Mbappe join Real two years ago, but Los Blancos were left in the dark over a deal and ended their pursuit for the time being.
A free transfer in the summer of 2022 appeared to be on the cards, but Real were left stunned when the 24-year-old extended his Parisiens contract until 2024 with the option of a further year.
However, Mbappe informed PSG over the summer that he does not intend to trigger his 12-month extension, meaning that he will be due to depart as a free agent at the end of the campaign as things stand.
Following Karim Benzema's surprise switch to Al-Ittihad, Mbappe was unsurprisingly tipped to finally complete his switch to the Bernabeu this summer, as PSG desperately tried to move him on.
Jude Bellingham's big-money arrival and Joselu's loan signing from Espanyol meant that a move for Mbappe was not feasible, though, and the France international was eventually re-integrated into Luis Enrique's squad.
A renewal at the Parc des Princes has not been completely ruled out, but all the signs are pointing towards Mbappe finally making the switch to the Spanish capital once his PSG deal runs out.
However, the La Liga giants have been forced to release a statement lashing out at "completely false" reports claiming that talks are already underway to bring the World Cup winner to Spain.
"Given the information recently issued and published by different media outlets, in which there is speculation about alleged negotiations between the player Kylian Mbappe and our club, Real Madrid CF would like to state that this information is completely false and that no such negotiations have taken place with a player who belongs to PSG," a statement read.
Nevertheless, should Mbappe continue to run down the final months of his PSG deal, the 24-year-old will be allowed to discuss pre-contract terms with Real Madrid from January onwards.
While Los Blancos are still seen as the frontrunners, a recent report claimed that Liverpool - who lost out to PSG for his signature in 2017 - are hopeful of a sensational free transfer ahead of the 2024-25 season.
Clubs in Saudi Arabia could also be keeping tabs on the situation after Al-Hilal agreed a €300m (£260.2m) deal with PSG, but Mbappe had no interest in leaving Europe at this point in his career.
The attacker has continued to shine on the field despite the ongoing speculation, coming up with 12 goals from as many games in the 2023-24 campaign so far, with 10 strikes in Ligue 1 and a further two in the Champions League.
Mbappe's total for Les Parisiens now stands at 224 strikes and 99 assists from 273 outings, surpassing Edinson Cavani to become his club's all-time leading goalscorer in March.