Earlier, state media said the military had recaptured all of the territory it had lost in Sunday's rebel assault.
Syrian President Bashar al-Assad and his army, along with allied Russian, Iranian and Shi'ite militia forces, have put rebels on the back foot with a steady succession of military victories over the past 18 months, including around Damascus.
Rebels detonated two large vehicle bombs at 5:20am on Sunday close to the Jobar neighborhood.
The Observatory, a British-based war monitor that collects information from a network of sources across Syria, said heavy fighting continued and that the army had unleashed more than 500 air raids and artillery strikes.
Sunday's fighting centered on a government-held intersection of two besieged opposition enclaves, the Jobar and Qaboun neighborhoods.
Sniper fire and air strikes were heard across the city on Sunday as civilians cowered inside their homes and schools announced they would close because of the violence. Control of Jobar - which has been a battleground for more than two years - is divided between rebels and allied jihadists and government forces.
But with Sunday's attack, Abdel Rahman said, "rebels have shifted from a defensive position in Jobar to an offensive one". The ultraconservative Ahrar al-Sham rebel faction said fighters had liberated the area.
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The source also said that the army units took control of Abbasiyeen area, north of Jobar as well as the power plant, textile industry and the surrounding buildings.
"These are not intermittent clashes - these are ongoing attempts to advance", he said.
The Observatory said fighting was still underway Monday.
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. United Nations Special Envoy for Syria Staffan de Mistura estimates that over 400,000 people have been killed in the conflict.
After a crackdown, the uprising became an all-out war that has drawn in world powers on almost all sides.
"The military operations north of Jobar targeted the areas from which the terrorists set out, and a large number of them were killed", it said.
Syria has been gripped by foreign-backed militancy since March 2011.