Forces loyal to President Bashar al-Assad launched the assault earlier this month with waves of air strikes, artillery fire and rocket attacks that are reported to have left more than 640 civilians dead.
Under growing international pressure to end the bloodshed, regime backer Russia last week announced daily five-hour "humanitarian pauses" in the enclave. But while the air campaign has eased, fighting has intensified on the ground.
Rami Abdel Rahman, the head of the Britain-based Observatory, said at least 12 regime fighters had been killed in two areas, Al-Rihan and Shifoniya, in overnight clashes with the Jaish al-Islam rebel group.
Jaish al-Islam shares control of rebel-held parts of Eastern Ghouta with Faylaq al-Rahman and Ahrar al-Sham. Damascus and Moscow say they are trying to clear the area of "terrorists".
Hamza Bayraqdar, a spokesman for Jaish al-Islam, said on Twitter that the group's forces had launched "surprise attacks" against regime positions.
- Hundreds flee -
An AFP correspondent inside Eastern Ghouta saw hundreds of civilians on Sunday fleeing from the town of Beit Sawa in the southeast of the enclave.
On Saturday, 18 civilians, including three children, were killed in regime bombardment of Eastern Ghouta, according to the Observatory.
At least 76 pro-regime fighters and 43 rebels from Jaish al-Islam have also been killed in clashes since February 25, it says.
Encircled by regime-controlled territory and unable or unwilling to flee, Eastern Ghouta's 400,000 residents have in recent weeks suffered one of the most ferocious assaults of Syria's civil war.
Under siege since 2013, they had already been facing severe shortages of food and medicine. The region's over-burdened medical workers have been struggling to cope with the rising number of wounded.
But trucks loaded with aid have so far been unable to enter the enclave, according to the UN.
Moscow has offered safe passage to non-combatants wishing to leave the enclave during the pause, but no Syrian civilians have left since the first break in fighting took effect on Tuesday, the Observatory says.
The Russian military said in a statement that no civilians exited via the established corridor on Saturday.
- 'Simply unacceptable' -
French President Emmanuel Macron and UN Secretary General Antonio Guterres expressed "grave concern" about the humanitarian situation in the area in a telephone conversation late Saturday.
Macron was set to speak by telephone Sunday with President Hassan Rouhani of Iran, another key supporter of the Syrian regime.
The UN's regional humanitarian coordinator for Syria, Panos Moumtzis, sounded the alarm on Sunday over the increase in violence.
"This collective punishment of civilians is simply unacceptable."
As Syria's conflict approaches its seventh anniversary, Assad's forces, heavily backed by Russia, have retaken most of the territory once lost to rebels.