According to the Manchester Evening News, the Spanish capital club are keen on securing the services of Mbappe. The 18-year-old scored 26 goals and registered eight assists across all competitions as Monaco went on to win the title for the first time since 2000.
Real are not the only club interested in signing the France international. United's local rivals Manchester City are also believed to be in the race in landing the teenager. Pep Guardiola's side have already made an initial bid of £143m ($186.2m) for the attacker.
The 12-time European champions are keen on bringing Mbappe to Santiago Bernabeu in the summer transfer window. They have already allowed James Rodriguez and Alvaro Morata to leave the club as they joined Bayern Munich and Chelsea respectively.
Zinedine Zidane admitted that he has not urged the club to bring in a player to replace Morata at Real. However, the La Liga winners are keen on beating City in securing Mbappe's signature and they need to raise funds to land the forward.
Real will sanction Bale's sale to United and will use the funds from the Welshman's sale to the Red Devils in signing the Monaco man. The 20-time English champions manager Jose Mourinho admitted that United's pursuit of Bale was long over after the player decided to stay with his current employers.
The situation is likely to change as Real look to sanction Bale's sale to fund Mbappe's transfer. The Portuguese tactician has publicly expressed his desire to sign an attacking player, despite signing Romelu Lukaku earlier in the window.
The possibility of Bale leaving Real is also being discussed in Spain. According to Cadena Ser, Real and Mbappe have agreed that four players – Mbappe, Karim Benzema, Cristiano Ronaldo and Bale – can be at the club at the same time.
The French striker wants assurances of regular playing time under Zidane and will only move to Spain if one of Bale, Ronaldo or Benzema is sold. The former Tottenham Hotspur star could be the one who would be allowed to leave this summer.
Mbappe has also warned Real that if they fail to sign him this summer, he will not join them a year later. Real are looking to agree personal terms with the player as they believe it will make their task easier in agreeing a fee with Monaco.
Should Bale decide to leave Real for United, he is likely to become the latter's third summer signing. Mourinho's side have already completed deals for Victor Lindelof and Lukaku as the two arrived at Old Trafford from Benfica and Everton respectively.