Rebels have tried to slow the advance with an attempted counter-offensive, but President Bashar al-Assad's forces steamrolled their efforts.
Government troops and allied militia have recaptured half of the besieged region in a blistering assault launched on Feb 18.
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.
At least 20 civilians - including four children - were killed in Douma on Saturday, in addition to 17 civilians in other battlefront towns, said the Observatory.
A negotiator from Hammuriyeh said a "civilian delegation" wanted to hold talks with the government to end the fighting in the area.
The government says its forces now control half the enclave.
Syria's war, which enters its eight-year next week, has killed more than 340,000 people since it erupted in 2011 with the brutal repression of anti-government protests.
Syrian army gains ground in intensified Ghouta assault
Takfiri groups in eastern Ghouta deny this, but eyewitnesses on Friday saw gunfire and mortar fire from inside the rebel territory near one of the crossing points. They encompass multiple factions, including jihadists, and in-fighting between them has led to past losses of ground to the Syrian government.
The council is to meet on Monday to hear a report from UN Secretary General Antonio Guterres and try to plod on with a bid to turn the ceasefire voted in NY into a reality on the ground.
Deputy Foreign and Expatriates Minister Fayssal Mikdad warned that well-known parties are leveling accusations against Syria regarding the use of chemical weapons as a result of the victories achieved by the Syrian Arab army in the fight against terrorism, SANA reports.
In January, Turkey began an offensive to oust the Kurdish YPG militia from the Afrin region in northern Syria, near the Turkish border.
Turkey regards the YPG as a terrorist group and says it is linked to the banned Kurdistan Workers' Party, which has fought an insurgency for Kurdish autonomy in Turkey for three decades.
Local doctors said that between 49 and 65 people had died on Friday, and the near-ceaseless shelling that started on 19 February resumed on Saturday morning.
Should we go to year-round daylight saving time?
An estimated 78 countries across the world observe Daylight Savings Time , however, Russia, India, China and Japan do not follow. A search on Change.org returns 1,466 petitions mentioning daylight saving time , most of which appear to support abolishing it.