Hamza Hassan, a surgeon working at one of the hospitals in eastern Ghouta, said that staff was overwhelmed with chlorine odour and that he treated 29 children with breathing problems.
The two largest groups are Jaish al-Islam and its rival Faylaq al-Rahman.
"It has been agreed that they be moved to Idlib at their request", the statement added.
Residents say they fear government retribution if they were to try and flee the area, or the possibility of being redrafted into the Syrian military and sent to fight on the frontline.
The aid was delivered with helicopters hovering overhead and warplanes targeting areas outside Douma, a correspondent in the town said.
"We are not cooperating with the UNHRC as it is politicized and propagating the policy of western states targeting Syria", Syrian Deputy Foreign Minister Faisal Mekdad told reporters.
It showed footage of people getting onto a bus. Residents have spent weeks living underground in bomb shelters, suffering food shortages and a lack of medical supplies, with doctors saying they are overwhelmed by the number of wounded.
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.More news: Prince Harry and Meghan Markle meet students on International Women's Day
The main groups had said on February 27 they would be willing to expel jihadist fighters from the enclave as soon as a United Nations ceasefire took effect.
Regime forces have advanced rapidly through farmland in the enclave since last week, according to the Observatory, taking control of more than half of the territory from the armed opposition.
The UN has called it "hell on earth".
A tribal leader said more than 300 civilians from the areas of Kafr Batna, Saqba and Hammuriyeh wanted to leave.
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.
But the government of President Bashar al-Assad has vowed to continue military operations, as it regards nearly all rebel groups as terrorists.
In January, Turkey began an offensive to oust the Kurdish YPG militia from the Afrin region in northern Syria, near the Turkish border.