De Rossi suffered the calf injury in Roma's 2-0 win over Cagliari at the weekend, forcing his departure in the 81st minute.
Considered one of Roma's most influential midfield players, the Italy international is now considered a major doubt for next week's Group E clash with City at the Etihad Stadium.
A statement from Roma said: "Daniele De Rossi underwent tests today that revealed a first-degree lesion to his left soleus muscle.
"He has already begun treatment on the injury."
Roma, who began their Champions League campaign with a 5-1 win over CSKA Moscow, sit top of the Italian top flight thanks to superior goals difference on Juventus ahead of a round of midweek fixtures.