The Bernabeu outfit have prioritised a move for the Spanish goalkeeper this summer.
Chelsea's Thibaut Courtois remains on Zinedine Zidane's radar, but De Gea is wanted to replace Keylor Navas.
As a result, Madrid are ready to break the world-record fee for a keeper and bid £60million, according to The Sun.
A Madrid source told the paper: “De Gea is the top priority.
“We expect him to come this summer.”
Florentino Perez is ready to spend big at Real Madrid to land Zinedine Zidane's top targets
Madrid president Florentino Perez is desperate to get a deal done for De Gea
The 26-year-old almost made the switch two years ago before complications on deadline day saw a transfer fall through.
De Gea still has two years remaining on his current contract, meaning Madrid will now have to spend big.
The fee would amount to almost double Gianluigi Buffon's £32m move to Juventus in 2001.
But Madrid president Florentino Perez is desperate to get a deal done for De Gea.
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And the La Liga giants will apparently try to conclude proceedings quickly with Manchester United rather than suffer another drawn out summer saga.
Madrid are intent on avoiding further embarrassment, following the mess over De Gea in 2015.
However, if United refuse to do business, with Jose Mourinho having recently warned his star shot-stopper to concentrate on matters at hand, Chelsea's Courtois will be targeted.
The ex-Atletico Madrid man remains Madrid's alternative and is considered a fail-safe option at the Bernabeu.