Eastern Ghouta has been besieged for years after many of its residents joined the initial protests against Assad's rule in 2011 that triggered the slide into civil war.
Turkey launched a military offensive into the border enclave on January 20 to drive out Syrian Kurdish forces that it considers to be "terrorists" and an extension of Kurdish rebels fighting inside Turkey.
During the army's offensive, more than 1,100 civilians have died, the UN Office of Humanitarian Affairs says.More news: Traders Purchase High Volume of Call Options on Express (EXPR)
Yasser Delwan, a member of the largest rebel group in the besieged region, confirmed that a group of patients had left under a deal for a medical evacuation with the Syrian government's key ally Russian Federation.
"We urge Russian Federation to stop creating distractions and compel the Assad regime to stop brutalizing innocent Syrian citizens and allow much-needed aid to reach the people of East Ghouta and other remote areas", Pentagon spokesman Eric Pahon said. "The Security Council failed to act". He also said the terrorist groups have targeted civilians in Damascus with more than 2499 mortar and rocket shells causing the martyrdom of 70 civilians and the injury of 556 others.
Panic is spreading in the town as the Turkish forces approach, and some civilians have come under fire when they have tried to leave, according to residents and Syrian Kurdish officials.
Afrin is already suffering from water and power shortages after Turkish forces cut supplies from the countryside, according to the Britain-based Syrian Observatory for Human Rights.
State television said the first group comprised about 35 people and they would go to a shelter on the outskirts of the nearby Syrian capital Damascus.More news: Slovak prime minister offers to resign to 'solve the political crisis'
"It is not the path we prefer, but it is a path we have demonstrated we will take, and we are prepared to take again", Haley told the council.
If the warring parties continue to ignore the truce, Haley said, the United States is prepared to take action on its own.
Redur Xelil, head of foreign relations for the Syrian Democratic Forces, a USA -backed bloc including the YPG, accused Turkish forces of carrying out "demographic change" by settling Turkmen and Arab families in Kurdish areas captured in Afrin. I refuse to lose my hope to see Syria rising from the ashes, he said, "To see a united, democratic Syria able to avoid fragmentation and sectarianism with its sovereignty and territorial integrity respected and to see a Syrian people able to freely decide their future and choose their political leadership". The US forces neither hold United Nations consent nor invitation from Syrian government which is considered to be violation of its territory.
In the regime-controlled area of Dweir on Monday, Rima Sheikh said she was devastated to have left a daughter in Eastern Ghouta. Recent government advances have cleaved eastern Ghouta into a northern and southern pocket.
Haley accused Syria and its allies Russian Federation and Iran of exploiting such "loopholes" to "continue starving and pummeling hundreds of thousands of innocent Syrian civilians" in Ghouta, the last opposition-controlled pocket near Damascus.More news: Activision Blizzard (ATVI) Earning Somewhat Positive News Coverage, Study Shows