Dybala joined Jose Mourinho's side on a three-year contract on Wednesday after his contract at Juventus expired.
He has opted for the number 21 jersey despite the number 10 jersey being available.
As one of the most gifted playmakers in Serie A, it would have only been natural for Dybala to take over the 10 jersey, which has remained vacant since Totti's retirement in 2017.
Totti spent his entire club career with Roma, becoming the club's all-time top scorer with 307 goals in 786 appearances; also a club record.
Dybala says he was offered the chance to take over Totti's famous number but politely turned it down.
"I spoke to the general manager and he asked if I wanted to take number 10, which of course is so significant here because of everything Francesco Totti did," Dybala said in a club interview.
"I think that shirt and that number should remain his because of what he means to the city and the fans. I thanked him because wearing a shirt like that requires a lot of respect and responsibility.
"I still see it as his shirt. Maybe I'll wear it one day but for now I'm happy to have number 21, which is the number I wore when I first started to be successful. Hopefully I'll now start being successful here too."
Dybala spent seven seasons with Juventus, winning five Serie A titles and four Coppa Italia trophies.
The 28-year-old scored 115 goals in 293 appearances for the Old Lady, making him Juventus' third all-time top scorer among foreigners to have played for the club.
