The Syrian government forces on Saturday captured the strategic town of Mesraba and encircled the city of Harasta in Damascus' Eastern Ghouta countryside, according to War Media, the media wing of the Syrian army.
It showed footage of people getting onto a bus.
"If I thought about leaving Ghouta I would worry about the regime because there are no guarantees: and, second, I would worry about Jaish al-Islam, because they won't let us leave", said one activist in Douma.
In a statement on Twitter on Friday, Jaish al-Islam, one of the main factions in eastern Ghouta, said the decision had been taken in consultation with the United Nations, a number of worldwide parties and civil society representatives from eastern Ghouta.
Syrian state news agency SANA said dozens of civilians were expected to leave Eastern Ghouta on Friday.
Rebels have tried to slow the advance with an attempted counter-offensive, but President Bashar al-Assad's forces steamrolled their efforts.
Syrian state TV reported that the army is close to cutting the enclave in two.
But the group represents only a small portion of the insurgent presence in the enclave, and both Jaish al-Islam and Failaq al-Rahman have said they would not negotiate a similar deal for themselves.
Aid agencies said Syrian authorities removed basic health supplies, including trauma and surgical kits and insulin, from the convoys before they set off.
Bilal Abu Salah, a resident of Douma, said shortages were causing great hardship.
Doctors Without Borders also estimates more than 4,800 people have been injured since pro-government forces intensified attacks on the area in February.
Insurgent groups in eastern Ghouta deny this, but a Reuters witness on Friday saw gunfire and mortar fire from inside the rebel territory near one of the crossing points.
Meanwhile, the Russian defense ministry extended an offer for armed rebels and their families not just civilians to leave eastern Ghouta through a safe corridor set up earlier for civilians, though none have left. It accused the government of using "poison gas".
"Living conditions are harsh".
"I don't want to leave, but I don't want any harm to happen to my family", said Douma resident Abu Ahmad al-Ghoutani, who said he has two children. The Observatory has reported protests in one village to demand an end to the bombardment and the departure of rebels.
"The bomb shelters and basements are full, and people are sleeping in the streets and in public gardens", the statement said.
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