"The vast majority of people thought he would have gone if we went down but he hasn't and he wants to see the job through.
"He now has a job to get us back into the Premier League again."
Coppell, who has been in charge of Reading for nearly five years and has 12 months remaining on his current contract, guided them to the top flight two seasons ago and to a magnificent eighth-place finish last year.
But Reading were consigned to the Championship on the final day of the current campaign, with Fulham's victory at Portsmouth rendering the Royals' victory over Derby meaningless.
Nevertheless, Coppell has remained a favourite with the fans and he revealed that their support did have some influence on his decision.
"The reaction of the supporters had a bearing on my decision, as did my relationship with the chairman and everyone at the club," he said.
"I'm now looking forward to the new season and the significant challenge of trying to regain our place in the Premier League."
Reading have also announced they are cutting season ticket and match ticket prices for next season.
The vast majority of adult season tickets have been cut by £120, while an adult match ticket has fallen to £23 in most areas, down from the 2007/8 maximum of £37.
"With Steve Coppell once again at the helm, we anticipate an exciting and memorable season," added Madejski.
"We hope to have as many Reading fans right behind us as we aim to win promotion back to the Premier League."
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