Casillas has been blamed by some fans for Real's woes this season, when they failed to win any of the three major trophies, while arch rivals Barcelona are on the verge of a treble of Champions League and Spanish league and Cup titles.
The Spain international has suggested he wants to see out his contract, which runs until the end of the 2016-17 season, but he has been visibly irritated by his treatment at the Bernabeu in recent weeks.
While some sections of fans clearly back him, others have roundly whistled him, along with coach Carlo Ancelotti and Wales winger Gareth Bale.
Madrid-based sports dailies Marca and As both reported on Wednesday that Real had let Casillas know the club wants him to stay on next season.
That did not mean, however, that Real would abandon their efforts to lure the 24-year-old De Gea, second choice for Spain behind Casillas, from Manchester United in the close season, they added.
If De Gea does move back to Spain, Costa Rica keeper Keylor Navas, who joined Real after shining at last year's World Cup finals in Brazil, would look to move on, reports said.
Casillas made his first-team debut for Real in September 1999 and has made 724 appearances in 16 seasons, winning a host of trophies including three Champions League crowns and five La Liga titles.
(Reporting by Iain Rogers, editing by Patrick Johnston)
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