Rebels and allied jihadists, led by former Al-Qaeda affiliate Fateh al-Sham Front, attacked government positions in the Jobar district and advanced into the neighbouring Abbasid Square area, the Syrian Observatory for Human Rights monitor said.
"They targeted government forces with two car bombs and several suicide attackers," Observatory head Rami Abdel Rahman said.
The Britain-based Observatory had no immediate information on casualties from the clashes.
Control of Jobar -- which has been a battleground district for more than two years -- is divided between rebels and allied jihadists on one side, and government forces on the other.
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It aired footage from Abbasid Square, typically buzzing with activity but now empty except for the sound of shelling.
AFP correspondents in Damascus said army units had sealed off the routes into Abbasid Square and explosions could be heard across the city.
Several schools announced they would close through Monday, and many civilians cowered inside in fear of stray bullets and shelling.
According to the Observatory, the Islamist Faylaq al-Rahman rebel group and the Fateh al-Sham Front -- known as Al-Nusra Front before it broke ties with Al-Qaeda -- were present in Jobar.
Government forces have long sought to push the rebels out of the district because of its proximity to the city centre in Damascus.
But with today's attack, Abdel Rahman said, "rebels have shifted from a defensive position in Jobar to an offensive one".
"These are not intermittent clashes -- these are ongoing attempts to advance," he said.
The Observatory said rebels had launched the attack from Jobar as a way to relieve allied fighters in the nearby districts of Barzeh, Tishreen, and Qabun from government attacks.
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