Five children from one family were killed with their parents when their neighbourhood was bombarded, activists said.
"Since this morning the bombing has not stopped," Abu Jihad Darawi, a Ghouta resident, said. "I was in the street and 20 rockets hit houses around me. The people are living as if they are dead."
Around 400,000 civilians are trapped inside Eastern Ghouta, which is the last major opposition-held area around Damascus.
The suburb has been under siege by regime forces since 2013 and supplies of food are perilously low.
Pro-regime forces have been massing around Eastern Ghouta since the weekend and the intensified bombardment has led many analysts to believe that Assad's fighters are preparing for a full-scale attack to recapture the area.
"Ghouta is of major symbolic importance to the regime because it is so close to Damascus and it matters for economic and security reasons," said Ibrahim al-Assil, a fellow at the Middle East Institute.
"If the regime takes it, they can tell their supporters that the war is almost over and the opposition only controls some pockets in the north, east and south but not in what the regime calls useful Syria in the west."
The regime will likely try to repeat in Eastern Ghouta the model it implemented during Aleppo and other sieges.
After bombarding and attacking the area, it may offer rebel fighters and civilians a deal, where they are allowed to leave for an opposition-held province in the north in return for abandoning their territory.
But that model is likely to be complicated by the size of the Eastern Ghouta pocket, where hundreds of thousands of people would need to be evacuated.
Separately, Syrian state media said that pro-regime forces were preparing to move towards the Kurdish-held pocket of Afrin in northern Syria, where Kurdish fighters are battling a Turkish incursion.
Syrian state media said the regime fighters would side with Kurdish troops from the People's Protection Units (YPG) and "join the resistance against the Turkish aggression".
Mevlut Cavusoglu, Turkey's Foreign Minister, warned the regime not to take the side of the Kurds in Afrin.
"If the regime is entering Afrin to oust the YPG, there is no problem. But if they are entering to protect the YPG, then no one can stop us," he said.
Turkey and the Syrian regime have exchanged threats and angry words for years but have not deliberately confronted each other on the battlefield.
Assil said it was likely that a deal would be negotiated before there was outright combat.
"Everyone is interested in reaching a deal: the regime, the Kurds, the Turks, the Russians," he said.