They moved tanks to defend their lines yesterday as their opponents struck targets elsewhere in the capital.
Residents and activists said the army sent armoured reinforcements to Jobar – a Sunni Muslim district adjacent to the city's landmark Abbasid Square – after rebels took over a loyalist position in the area, the third since fighters pushed into Jobar last week.
However, Mr Assad's forces were entrenched in the centre.
An opposition activist in Damascus said: "The main battle is taking place in Jobar. The rebels appear to be advancing in the eastern sector. But the centre of Damascus is crisscrossed with concrete barriers and security is deployed everywhere.
"We cannot say the rebels have a real active presence in the city centre."
He said the Army seemed to be under so much pressure in Jobar it had moved tanks there from the south-western suburb of Daraya, near the highway to the Jordanian border, where it has been battling rebels for two months.
A video by a group known as Liwa al Islam, one of the opposition fighting units around Damascus, showed its members firing a rocket they claimed had a range of 35 miles, an apparent marked improvement in the arsenal of the opposition.
Sham News Network, an opposition group of media activists, said rebels overran an Army barracks in Jobar and had attacked a roadblock in the Afif district overnight. Afif is located near one of Mr Assad's presidential compounds in the foothills of Mount Qasioun, northwest of the city.
Activists also reported a mortar attack on a police station in the central Damascus district of Arnous.
It was not clear whether the mortars hit the target or if there were casualties.
One women in the western neighbourhood of Mezze said: "The situation is getting tough. For the first time we have been hearing mortars fall so close."
The Syrian military has been firing heavy artillery and rockets from Qasioun at Jobar and a series of Sunni Muslim districts at the forefront of the uprising against the regime.
Mr Assad's core forces, from his minority Alawite sect – an offshoot of Shi'ite Islam – are based on Qasioun.
Their main supply line to coastal bases passes near the contested central city of Homs, 90 miles north of Damascus.
Activists yesterday reported clashes in the Qalamoun region on the Damascus-Homs highway, near an army base from where Scud missiles have been fired at rebel-held territory.
Meanwhile, activists said rebels had taken control of the country's biggest hydro-electric dam on the Euphrates River.
The Syrian Observatory for Human Rights and other activists said Islamist fighters seized the dam entrances, although gunmen had not entered the main operations room and the dam continued to function.
They had earlier swept through the nearby town of Tabqa, renamed al Thawra (Revolution) by the country's rulers. A statue of Mr Assad's father, the late President Hafez al Assad, was set on fire in the town.
Other video footage showed what activists said was an abandoned Air Force Security base next to the dam and army installations inside the town.
Rami Abdulrahman of the British-based Syrian Observatory described the collapse of Mr Assad's forces in Tabqa and around the dam as one of the president's biggest strategic setbacks in the 22-month-old Syrian uprising.