Another 21 were injured in the shelling of the Bab Touma and al-Shaghour districts of the old city, the source said.
The Syrian Observatory for Human Rights said 11 were injured in Bab Touma and seven elsewhere in the Old City. Some are in critical condition.
Rebel factions in the besieged enclave of Eastern Ghouta have escalated their shelling of government-controlled areas in and around Damascus since November, the Britain-based war monitor said.
Eastern Ghouta is the last major rebel-held pocket near the capital.
A man is seen near the remains of a rocket in Douma, Eastern Ghouta in Damascus, Syria
Nine Syrian civilians were killed on Monday by insurgent shelling of the old quarter of Damascus, Syrian state television said, citing a source from the capital's police.
