The Syrian Observatory for Human Rights (SOHR) said the victims comprised 26 security personnel and 21 militants, including two who blew themselves up in suicide car bombings, Efe news reported.
Several factions took part in the clashes, including Tahrir al-Sham (the armed alliance that includes the former Syrian branch of Al Qaeda), Ahrar al-Sham and al-Rahman Legion.
Tahrir al-Sham announced on Sunday the beginning of an offensive in the Damascene neighbourhood of Jobar, in the northeast and near the old part of the city.
Warplanes carried out 10 strikes on Monday in Jobar, which is also the target of artillery shots.
On March 15, at least 31 people were killed in a suicide bomb attack at the main court complex in the centre of Damascus, the BBC reported.
Later that day, another suicide bomber attacked a restaurant in the district of Rabweh, injuring more than 20 people.
March 15 marked the sixth anniversary of the beginning of the uprising against President Bashar al-Assad, which has caused more than 321,000 deaths.