According to Antony Kastrinakis in The Sun, Los Blancos will bid £60 million for De Gea at the end of the season as they "desperately want" him at the Santiago Bernabeu.
Per Kastrinakis, a Madrid source said: "De Gea is the top priority. We expect him to come this summer."
BBC News' Neil Henderson provided The Sun's back-page splash:
The current record fee for a goalkeeper is the £32.6 million Juventus paid Parma for Gianluigi Buffon in 2001. De Gea became the second-most expensive when he moved to United from Atletico Madrid for an estimated £18.9 million in 2011, per BBC Sport.
De Gea, 26, has been linked with Los Blancos for some time and a deal for him to join Real fell through on transfer deadline day in 2015, per Sky Sports.
The Spanish giants' interest has seemingly never waned since, though, as De Gea has continued to prove himself as one of the best goalkeepers on the planet.
However, per Kastrinakis, Real president Florentino Perez has no interest in a protracted transfer saga around De Gea this summer, and the club will switch their focus to Chelsea stopper Thibaut Courtois if they cannot seal a quick deal for the Spain No. 1.
Again, given that Courtois, 24, is one of the best goalkeepers in Europe, Real have been linked with him before.
It was even reported back in December that a deal was close to being agreed, per Onda Cero (h/t Jose Felix Diaz in Marca).
As with De Gea and United, though, Chelsea are unlikely to be open to letting Courtois go as he is a vital part of manager Antonio Conte's starting XI and could continue in the role for more than a decade.