The Syrian Observatory for Human Rights, a British-based war monitor, said heavy fighting continued around the Jobar and al-Qaboun districts in the northeast of the city.
"The government and allied forces have retaken the initiative and are striking the groups that launched yesterday's assault", he told AFP news agency.
Control of Jobar has been divided between moderate rebels, government troops, and jihadists for more than two years.
Activists said opposition-held parts of Jobar had been targeted repeatedly by aircraft and artillery, amid fierce clashes along the frontlines.
Abdel Rahman said it was unclear whether government forces or their Russian allies were carrying out Monday's raids on Jobar.
"We understand that since daybreak there have been around 30 air strikes in [Jobar]", Tyab said, adding that there were reports of heavy rebel losses, although credible figures were not yet available.
The Abbasid Square is a gateway to the heart of Damascus, the seat of President Bashar al-Assad's government. It aired footage from Abbasid Square, typically buzzing with activity but now empty except for the sound of shelling.
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Military source said that army units launched a counterattack on Nusra Front terrorists in which it secured the factories region north of Jobar, Hezbollah's Military Media Center reported.
While the strategic objective of Sunday's offensive is unclear, observers say a possible goal was breaking the government's siege of Qabun.
"They targeted government forces with two vehicle bombs and several suicide attackers", Observatory head Rami Abdel Rahman said, adding that there was no immediate information on casualties from the clashes.
Also Monday, Syrian state TV and the Observatory reported that an airstrike in the country's north - said to be by the US -led coalition, which has been targeting the Islamic State group in Syria for more than two years - had left at least seven people dead, including five members of the same family.
The assault was led by Islamist groups including Ahrar al-Sham and the recently formed umbrella jihadist group Hayat Tahrir al-Sham.
Syria's conflict erupted in March 2011 with protests against Assad's rule but has evolved over the years into a complex civil war.