"We have reports that a shell that struck... Near the Sheraton Hotel in Umayyad Square injured two people... While reports emerged of five killed and 20 wounded when shells hit Abbasiyeen Square and (nearby) Qassaa," the Syrian Observatory for Human Rights said.
Among the dead was a singer of Christian religious songs, Observatory director Rami Abdel Rahman said.
Both Abbasiyeen and Umayyad squares have been targeted by intermittent rebel shelling in recent weeks, as violence has escalated on the outskirts of Damascus.