The club confirmed in a statement that Roma's final game of the season - at home to Parma - would be De Rossi's last for the club he joined as a teenager in 2000.
Roma chief executive Guido Fienga said the club only informed De Rossi on Monday of their decision not to renew his contract.
"I apologised to Daniele, on behalf of the club, for not speaking to him sooner - it has been a problematic season," Fienga told a news conference.
"The club is aware of the mistakes we've made, and we are reflecting on that. But we appreciate the way Daniele has accepted the decision."
Fienga offered Rossi a director's role at Roma but the former Italy international has decided to extend his playing career.
"I'm almost 36 and I know how football is, but I didn't want to distract from our European (qualification) race," De Rossi told reporters.
"I thank the directors for their offer - I respect it and appreciate it, but I still feel like a player."
"Eventually, I would like to be a coach. At this point I don't see myself becoming a director, but at Roma a role like that would have some meaning. I need to go away and study, to learn - it will be a long journey."
De Rossi won two Coppa Italia titles with Roma in 2007 and 2008, although the nearest he came to a Serie A crown was in 2010 when Roma led the table with four games to go but were overhauled by Jose Mourinho's Inter Milan.
"We will all shed tears when he pulls on the Giallorossi (yellow-and-red) shirt for the last time against Parma but we respect his desire to prolong his playing career - even if, at almost 36, it will be away from Rome," said club president James Pallotta.
"I'd like to thank Daniele for his incredible dedication to this club and ensure him that our doors will remain open for him to return to the club in a new role whenever he wants."
De Rossi also played 117 games for Italy, making him their fourth most-capped player, and came on as a second-half substitute in the 2006 World Cup final in which they beat France on penalties after a 1-1 draw.
De Rossi missed most of that tournament, however, as he was serving a four-match suspension for elbowing an opponent in their opening game against the United States.
He was also sent off nine times playing for AS Roma but there was far more to De Rossi than aggression, as he is famous for his vision, passing ability and eye for goal.
(Reporting by Brian Homewood; Additional reporting by Hardik Vyas in Bengaluru; Editing by Ed Osmond)
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