The activist, who goes by the name of Abu Raed because of security concerns, said the rebels are moving into the neighboring Al-Siryan Al-Jadideh, a Christian area.
“It was a surprise,” Abu Raed said. “It was fast progress and in an unexpected direction.”
Abu Raed said he had no information on people killed or wounded in the fighting, but the Britain-based Syrian Observatory for Human Rights said nine people were killed and 15 wounded when mortars fell on Ashrafiyeh. The activist group said it was unclear who fired them.
Activists also reported heavy shelling and clashes in various rebel districts outside of Damascus, the capital.
Also on Thursday, the deputy head of the United Nations, Jan Eliasson, warned that there are no guarantees that a proposed Syrian cease-fire would hold, but he urged Syrian rebels and the regime in Damascus to observe it.
“We all have our eyes on the tragedy in Syria, and we pin our hopes now on the cease-fire that hopefully can take place,” he told reporters in Geneva.
Previous cease-fire missions have failed, in part because neither Mr. Assad nor rebels trying to topple him had an incentive to end their bloody war of attrition. Both sides believe they can still make gains on the battlefield even as they are locked in a stalemate, and neither has faith in negotiations on a political transition.
Mr. al-Dada said the regime could not be trusted.
“Our people have no truce. They have been subjected to massacres,” he said.
Associated Press writers Barbara Surk, Bassem Mroue and Karin Laub in Beirut and John Heilprin in Geneva contributed to this report.