The Manchester United star has been heavily linked with a move to the Spanish giants in recent months.
De Gea, 26, nearly joined Real in 2015 but a late fax ended any hope of a deal.
The Spain international instead went on to pen a new contract at Old Trafford.
However, Spanish news outlet Diario Gol say De Gea will end up at Real.
They claim the United stopper has an unwritten agreement to join the La Liga champions.
Real president Florentino Perez was desperate to sign De Gea this summer but Zinedine Zidane had other ideas.
The Real boss wanted to put faith in Keylor Navas, who impressed under Zidane last time out.
“David De Gea has an unwritten agreement to join Real Madrid”
Diario Gol say Navas knows he will leave the Bernabeu next summer and he’ll be replaced by De Gea.
United have had a fine start to the season under Jose Mourinho.
The Red Devils have won both of their Premier League clashes 4-0, with Romelu Lukaku and Paul Pogba shining for United.
United are back in action on Saturday when Mourinho’s men face Leicester at Old Trafford.