Spain's No1 has long been linked with a move to the Bernabeu, having come close to completing a switch in 2015.
While United managed to hold on to their star shot-stopper, another pursuit is ongoing for this summer.
Jose Mourinho has been identifying potential replacements such as Atletico Madrid's Jan Oblak in case he loses David De Gea to the Champions League winners.
And to avoid another saga like the one between the European giants two years ago, Madrid are pushing ahead.
Real Madrid are pushing hard to sign Manchester United goalkeeper David De Gea quickly
Neither De Gea or Madrid want a deal to be prolonged
According to Spanish newspaper Marca, neither De Gea or Madrid want a deal to be prolonged.
And both hope everything will be done this week.
The self-imposed time constraint shows Madrid president Florentino Perez is eager to land his man early in the transfer window.
But the biggest obstacle remains the fee to be agreed with Manchester United.
David De Gea transfer: Potential replacements for the Man Utd keeper Thu, June 1, 2017
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However, Marca add the final price is expected to be around £65million, which is what Mourinho's side have been demanding.
If the clubs do not come to an agreement quickly though, it is suggested the deal could collapse.
The Manchester United star already hinted at a goodbye to Old Trafford in his end of season message.
But with the clock said to be ticking over the conclusion of De Gea's move, it could be affected if it is dragged out much longer than this week.