Russia-backed government forces on February 18 launched an assault of the besieged region, pounding the area with air strikes, killing more than 630 civilians.
On February 25, clashes on the ground between the armed opposition and regime forces intensified on the edges of the enclave, the Britain-based Syrian Observatory for Human Rights monitor said.
Since Thursday, regime forces have advanced inside the region, retaking an area in the enclave's southeast, as well as another and two military bases in the south from main opposition group Jaish al-Islam.
Government fighters are trying to advance inside Eastern Ghouta to cut off the main town of Douma and its surroundings in the north of the enclave, as well as isolate the southeastern area of Al-Marj, Abdel Rahman said.
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The Observatory says "Russian advisors" are taking part in the battles alongside regime forces in Eastern Ghouta, backed by heavy air strikes and shelling.
Since February 25, 60 regime and allied fighters have been killed, while 34 rebels from Jaish al-Islam have also died, it said.
Eastern Ghouta's 400,000 residents have lived under regime siege since 2013, facing severe food and medicine shortages even before the latest offensive.
The ground offensive coincides with the start of a five-hour daily "humanitarian pause" announced by Russia and in force since Tuesday.
The bombardment has lessened but continued to claim lives since, and trucks loaded with desperately needed aid have remained unable to enter the enclave.
Moscow said it would provide safe passage to non-combatants wishing to leave the enclave during the pause, but no Syrian civilians have left since Tuesday, according to the Observatory.
More than 340,000 people have been killed and millions have fled their homes since the start of Syria's civil war in 2011 with the brutal repression of anti-government protests.