Both players are amongst some of best talents in world football, with a number of European football's big hitters being linked with them.
talkSPORT understands that the Champions League holders are confident they will win the hotly contested race to sign Bellingham.
Liverpool and Manchester City have also been linked with the 19-year-old, but Madrid are convinced they can complete a deal.
Bellingham has enjoyed a stunning rise in world football and could be joined by another superstar in the Spanish capital.
As reported by The Telegraph, Madrid are set to 'explore' a move for Paris Saint-Germain forward Mbappe as well.
The Frenchman has been a previous target of Los Blancos, who tried to sign him last season before he signed an eye-watering new deal with PSG.
However, they could return for the 24-year-old as PSG look set for a huge summer of change.
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Lionel Messi has been linked with a move to Saudi Arabia, and fans of the French giants chanted 'f*** off' to Neymar outside his home.
Mbappe departing would be another blow to PSG, who are 'determined' to keep hold of his services and build their team around him.
However, the final 12 months of his deal that runs to 2025 are said to be an optional year, so clubs could make offers in the summer.
One of those is Madrid, who will reignite their interest in Mbappe if they believe there is a chance he could be available amid all the uncertainty in Paris.
The report adds the club have already discussed deals for both him and Bellingham, and the duo would arrive as part of their new 'Galacticos' side.
Madrid could therefore be set for a record summer spending spree, one that would be backed up by a potential Champions League win.
They drew 1-1 with City in their semi-final first leg on Tuesday night, with Vinicius Jr's stunning strike being cancelled out by Kevin de Bruyne's equally impressive effort.
Madrid will travel to the Etihad Stadium next week for the second leg as they set their sights on what would be a second consecutive final.
Should they progress and beat either AC or Inter Milan in the Istanbul showcase on June 10.