Almost 400,000 people have been trapped in Eastern Ghouta since 2013 when insurgents first took over control and the government began its offensive to oust the rebels.
At least 976 civilians, including 208 children, have been killed in Eastern Ghouta since February 18, when al-Assad's forces started a major offensive against rebel-held areas there, according to the Observatory.
The two largest groups are Jaish al-Islam and its rival Faylaq al-Rahman.
The government-controlled Syrian Central Military Media said troops reached the centre of the town of Misraba, adding that opposition fighters are crumbling amid the offensive.
Late on Friday, a small number of fighters and their families from the former al Qaeda affiliate previously known as the Nusra Front left eastern Ghouta under such a deal.
A charity called for desperately needed medical supplies to be allowed into the enclave, where exhausted doctors and nurses have been struggling to treat hundreds of wounded with very little equipment.
Taking advantage of the relative quiet, 13 trucks loaded with 2,400 food parcels entered Eastern Ghouta on Friday morning, the International Committee of the Red Cross said.
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But this can be done only after permission from all family members and a medical board created to look into the case is received. Journalist Pinki Virani had fought for her right to die with dignity , which was ultimately denied by the Supreme Court .
The aid was delivered as helicopters hovered overhead and warplanes targeted areas outside Douma, according to an AFP correspondent.
In the face of the government's Russian-backed onslaught, Ghouta's main rebel groups had so far rejected Moscow-brokered offers to evacuate civilians or any of their own fighters.
The military gains have caused wide-scale internal displacement as civilians flee government advances toward areas in the territory still held by the rebels.
The violence has continued despite a United Nations security council resolution demanding a 30-day ceasefire and the delivery of humanitarian aid "without delay". The channel has been reporting since last week that rebels have prevented civilians from leaving.
A tribal leader said more than 300 civilians from the areas of Kafr Batna, Saqba and Hammuriyeh wanted to leave.
But the government of President Bashar al-Assad has vowed to continue military operations, as it regards nearly all rebel groups as terrorists.
Turkey-led rebels have been pressing an assault on the Kurdish enclave of Afrin since January 20.