De Rossi, 35, will leave at the end of the season.
"Today's a sad day," Totti wrote on Instagram.
“Today, another important chapter in Roma's history closes, but above all Rome... our Rome.
“Over the years we've been through so much together. We've grown together, becoming men first, then footballers and eventually parents.
“But we knew this moment would come at some point. I just want to tell you that you have been and always will be my brother.
“I wish you the best in everything you do... because I'm sure you'll do them great, like everything you've done so far.
“I love you, Dani."