Spanish TV station TVE claimed on Wednesday that Casillas has reached an agreement to leave the club this summer despite the Spanish goalkeeper having two years left on his contract.
Real have been strongly linked in recent months to acquire Manchester United goalkeeper De Gea.
De Gea, 24, is under contract at Old Trafford for another season but has failed to agree a new deal with the Red Devils.
Both goalkeepers are currently on international duty with Spain, who take on Costa Rica in an international friendly in Leon on Thursday.
Casillas, the Real and Spain captain, stated last month that he had no intention of leaving Real any time soon and would fight for a starting place with De Gea, should the former Atletico Madrid player’s potential move to the Bernabeu stadium materialise.
“I don’t see myself anywhere else next season,” the 34-year-old Casillas recently said. “I’ve already stated that I want to retire here.
“I would like to end my career at Real Madrid.
“They (the club) recently told me they want me to continue.
“I am a man of challenges and last year I had a big one and I overcame it.”
Casillas lost his starting place under former Real coach Jose Mourinho and also struggled at the World Cup, where Spain exited at the first hurdle last year.
Real signed goalkeeper Keylor Navas to a six-year deal last summer, and the Costa Rica international did make a handful of starts earlier in the 2014-15 campaign for the Spanish club, but Casillas regained his spot in the first XI.
Real did not win any silverware in 2014-15. They finished runners-up to Barcelona in the Primera Division and were knocked out of the Champions League in the semi-finals by Juventus.
Their disappointing end to the campaign led to the sacking of Carlo Ancelotti, who had guided Real to a record 10th European Cup in the 2013-14 campaign.
Real appointed Rafa Benitez as Ancelotti’s replacement earlier this month.
Casillas would have no shortage of offers, according to TVE, with the veteran goalkeeper having reportedly 10 offers on the table, including one from Turkish giants Fenerbahce.
A World Cup winner and two-time European champion with Spain, Casillas has clinched five Primera Division titles, three Champions League trophies and two Copa del Rey crowns since making his first-team debut in 1999.