According to monitoring group Syrian Observatory for Human Rights, government jets have launched as many as 30 air strikes to quell the infiltration bid by the rebels. "The opposition is trying to make some military gains that could leverage their talks with the Assad regime in Geneva".
The clashes centred on the city's Abasseen neighbourhood, a government-held area sandwiched between the besieged, opposition-held Jobar and Qaboun neighbourhoods.
Insurgent groups attacked government positions in the area, including with two auto bombs, in an attempt to storm their defence lines, the Britain-based Observatory said.
Government forces responded with almost a dozen air strikes on Jobar, he added.
Syrian state media reported that Syrian forces had repelled an attack by jiahdist fighters, who also used tunnels to attack government positions.
State television aired footage from Abbasid Square, typically buzzing with activity but now empty because the army had ordered residents to stay indoors.
Several tanks were seen entering east Damascus as reinforcements.
Schools announced they would be closed Monday for fear the shelling would continue. The opposition only has a few pockets of control around Damascus, including a part of Jobar.
SOHR chief Rami Abdel Rahman said the rebels were seeking to "link their territory in Jobar with Qabun to break the government siege there".
"These are not intermittent clashes - these are ongoing attempts to advance", he said.
Rebel sources said their attack on parts of Jobar, which they held for a time in 2013, was in response to their loss of ground in Qaboun and Barza, two other districts to the north.
But the state-run SANA news agency downplayed the recent events in Damascus "Army units thwarted the attack by the terrorists and isolated them in the area".
A suicide bomber last Wednesday killed at least 31 people in the main court complex, and 20 more were injured in another attack at a restaurant later in the day, marking the sixth anniversary of the uprising against President Bashar Assad.
An AFP photographer saw men, women and toddlers peeking out from behind curtains as the buses headed to Jarabulus, a town on the Syrian-Turkish border.
The United Nations has sponsored peace talks to end the conflict since 2012, but to no avail.
Tens of thousands of fighters, dissidents, and their family members in long-besieged areas have accepted exile to the country's rebel-held northwest, in what opposition figures have termed "forced displacement".