The Spanish international has nine months remaining on his contract at Old Trafford, the Mirror adds, but he will have to see out those final few months before moving to the Bernabeu.
A recent report from the Independent suggested that De Gea could be in talks over a new deal with the Red Devils, which would have been a shock twist after long looking like he was set to quit the club for a move back to his native Spain.
Real were in talks over a move for the talented 24-year-old, only for the transfer to collapse in bizarre fashion late on due to a delay in submitting all the correct paperwork, as reported at the time by BBC Sport.
Although Madrid signed Kiko Casilla as another option in goal during the summer, it seems likely that they could still be keen to land De Gea as a long-term replacement for departing club legend Iker Casillas, who ended his long association with Los Blancos to join Portuguese giants FC Porto.
This will likely come at the expense of United, whose fans will no doubt be disappointed to see this inevitable departure of such a quality shot-stopper, who was arguably their best player for much of last season.