The BBC's Martin Patience in Beirut says it is not clear whether Jaysh al-Islam's fighters want to give up their weapons, and possibly become a security force under government control, or be bussed to a rebel stronghold in northern Syria.
A Russian-brokered deal had been reported yesterday for fighters with Jaish al-Islam, the largest rebel group still in Ghouta, to leave the enclave's main town of Douma.
"Turkey will not hesitate to use initiative if terrorists in Manbij are not driven away", a statement on President Recep Tayyip Erdogan's presidential website last week read, warning the United States whose special forces there in the northern Syrian town continue to work with Kurdish-led groups.
The SANA news agency says two buses carrying the rebels left Douma on Monday morning, heading for Jarablus, a town in north Syria shared between rebels and Turkish forces.
"There are attempts to convince the hardline wing of Jaish al Islam not to obstruct the agreement with the Russians", said the head of the Britain-based monitor, Rami Abdel Rahman.
The deal over Douma would mark the end of a weeks-long push by Assad's forces to consolidate their control over eastern Ghouta, just outside the capital.
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Douma was one of the earliest centers of the anti-government demonstrations that swept through the country in March 2011. Syrian government forces responded by putting the town and other suburbs around Damascus under siege, bombing hospitals and residential areas, and blocking the entry of food and medical relief.
The most recent Syrian air and ground offensive on eastern Ghouta, supported Russia's military, killed at least 1,600 people, according to the Observatory. More than 120,000 people fled their homes under the five-week assault, seeking safety behind government lines, according to the Russian military.
Backed by Russia, Mr Assad's forces have scored a series of victories over rebel forces in recent years, often through campaigns of siege, aerial bombardment as well as ground offensives that have drawn widespread worldwide condemnation.
Turkish state-owned media on Sunday claimed the USA is bolstering its military presence in Syria's northern district of Manbij against the possibility of another incursion by Turkey following the fall of Afrin.
The unit of the powerful Lebanese Shia militia that has been fighting with the Syrian army said among the terms of the deal was setting up a Syrian government-approved local council to run the city's affairs after rebels withdraw.
"Coalition forces, in an advise, assist and accompany capacity with our partners, were conducting a mission to kill or capture a known (I.S.) member when they were struck by an improvised explosive device", the coalition said in a statement to the AP. The U.N. has estimated the material cost of the war damage at close to $250 billion.