Roma snapped up Argentina international Dybala on a three-year contract on Wednesday after his deal with Juventus expired.
Totti made 786 appearances and scored 307 goals in a 25-year career with the Giallorossi that came to an end at the conclusion of the 2016-17 season. The club has not had a number 10 since then.
Dybala was grateful to be offered the chance to wear the shirt but felt it should remain Totti's. However, he did not rule out donning it in future.
"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."
Jose Mourinho steered Roma to their first trophy since 2008 by winning the Europa Conference League last season and Dybala wants to help them push for more silverware.
The five-time Scudetto winner is looking forward to working alongside Mourinho, who he considers to be one of the all-time great coaches.
"First of all, I want to get into peak shape so that I'm on a par with the other players mentally and physically and help the team doing my job. I want to help this team to carry on winning, with the mentality they acquired last season thanks to the coach, and with my own experience.
"Of course, the World Cup is around the corner so I'll be doing my absolute best to help this team and reach the tournament in the best shape possible, then hopefully be in the mix for some silverware at the end of the season."
He added: "Everyone knows who Mourinho is, what he's achieved in the game and what he can do. It was an exciting conversation and we spoke for a long time. It was a very nice chat.
"I've been fortunate to play with some of the greatest players of all time and now I'll get to work with one of the greatest coaches of all time."
Roma appointed Mourinho last May and invested around €130million on new players – the biggest close-season outlay of any Serie A team.
Things started positively with Roma winning their first six competitive games of a season for just the third time in their history, but they have been victorious in just four of their subsequent 12 outings.
Mourinho became the first Giallorossi boss since Luis Enrique in 2011 to lose their maiden top-flight Derby della Capitale in September, while the following month he suffered the heaviest loss of his managerial career when Roma were hit for six by Bodo/Glimt in the Europa Conference League.
The next week his 43-game home unbeaten run in Serie A came to an end at the hands of Milan, while November started with a disappointing draw at home to Bodo/Glimt and a shock 3-2 loss at Venezia.
However, Totti does not believe Mourinho is at fault for Roma's slide in form and believes his pedigree is more than sufficient for a show of faith.
"If we think Roma's problem is Mourinho, it means we have everything wrong," Totti told Sky.
"Mourinho has won more than all the other Serie A coaches combined – hats off to him.
"You have to focus on him. He is a great coach and a great motivator. He knows how to manage the group, he knows what to say and what to do.
"The club and the fans must stick with him."
Despite their poor form, Roma sit sixth in Serie A and are just three points adrift of the top four.
The 34-year-old's Camp Nou contract is set to expire on July 1, leaving him free to move on from the club where he has spent the entirety of his 17-year professional career thus far.
Fortunately for Barca, that outcome looks unlikely, with recent reports suggesting that the Argentina great is on the verge of agreeing fresh terms.
Should Messi put pen to paper on that extension, it will increase the possibility of him joining the ranks of players who spent their whole careers at just one club.
Here, we take a look at five of football's most celebrated one-club men.
Paolo Maldini (AC Milan)
Indisputably one of the greatest defenders of all-time, Maldini made his Milan debut as a 16-year-old in 1985 and spent the next quarter of a century at San Siro, winning seven Serie A titles and five Champions Leagues.
Ryan Giggs (Manchester United)
Like Maldini, Giggs progressed from prodigious young talent to distinguished elder statesman as he represented United from 1990 to 2014. Two Champions Leagues and 13 Premier League titles sit among an astonishing 34 honours amassed by Giggs across 963 senior United appearances.
Francesco Totti (Roma)
Totti bid an emotional farewell to Roma at the end of the 2016-17 campaign after 786 competitive appearances and 307 goals – matching Maldini's record of appearing in 25 Serie A seasons.
Carles Puyol (Barcelona)
A long-time team-mate of Messi's and the captain of Pep Guardiola's mesmeric treble winners of 2008-09, Puyol won six LaLiga titles and three Champions Leagues. With 593 appearances for Barcelona to his name, the defender sits fifth on the club's all-time list behind Sergio Busquets, Andres Iniesta, Xavi, and the outright leader Messi on 778.
Matthew Le Tissier (Southampton)
A mercurial forward with a knack for scoring improbably audacious goals, Le Tissier was regularly linked with the leading lights of English football during his 1990s heyday. He remained loyal to boyhood club Southampton to cement icon status on the south coast, although a mere eight appearances for England provides a talking point about what might have been possible had he not resisted offers from bigger clubs.
