Thousands of fighters have already evacuated the suburbs with their family members and other civilians, but the Army of Islam said it was determined to stay in its stronghold, Douma, the largest town in eastern Ghouta.
Russian-backed Syrian troops and allied militia have waged an offensive since February 18 in which they have recaptured more than 90 per cent of Ghouta, a one-time rebel bastion on the edge of Damascus.
In the previous two batches, on last Saturday and Sunday respectively, a total of 6,416 people, including rebels and their families, evacuated the same areas and moved to the rebel-held areas in the northwestern province of Idlib.
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They have usually begun with the military encirclement of an area, followed by bombardment and a ground operation before a deal is reached.
After the full evacuation of the Failaq al-Rahman rebels, only the Douma district in Eastern Ghouta would still be controlled by the rebels. At least 1,600 civilians have died in the carnage, and 5,000 more have been wounded, according to the United Kingdom -based observatory, which gathers information from a network of activists on the ground.
Devastating air strikes and artillery fire have reduced large parts of Ghouta to ruins, forcing their residents to abandon them.
Some 110,000 people have fled into government-held territory, a lot of them on foot or motorbike, through corridors opened up by the army and its Russian allies, according to state media.
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Late last week, the Russian Defense Ministry said that civilians have continued to leave Eastern Ghouta via the humanitarian corridors, and that more than 108,000 people have already fled the area since the introduction of daily humanitarian pauses.
Under the second deal, opposition fighters from Faylaq al-Rahman are being given safe passage to the northwest, along with thousands of civilians, in exchange for releasing prisoners they were holding.
Al-Watan quoted Syrian legislator Mohammed Kheir Seiryoul, who is originally from Douma, as saying that the understanding could lead to an agreement to dissolve the Army of Islam. "This sped up the regime's advance".
More than 120,000 crossed the front lines for safety, the Russian military said Tuesday.
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It said a "preliminary understanding" had been reached that would see the "dissolution of Jaish al-Islam, the handover of its heavy weapons and the return of state institutions to the city".