Jaish al Islam, most of whose members are drawn from the area, has been defending the city of Douma against a month-long onslaught by government forces.
If confirmed, its surrender or departure for rebel-held areas in northwest Syria would herald an end to large-scale conflict in the enclave.
State television and Syrian newspapers said information indicated that, under the deal, the group would hand over heavy and mid-sized weapons and acknowledge the restoration of state sovereignty over Douma.
Jaish al Islam had no immediate comment on the reports.
The government lost control of Douma, the largest urban centre in the eastern suburbs of Damascus remaining in insurgent hands, in the early phase of Syria's civil war, which is in its eighth year.
If confirmed, the agreement with Jaish al Islam would mirror others reached by insurgents following heavy bombing of areas under their control by government forces and Russian jets that has led to hundreds of casualties among civilians.
Late on Saturday, negotiators reached a deal to evacuate wounded Jaish al Islam civilians and fighters to Idlib in the northwest.
A group of stranded fighters from another insurgent group,
Failaq al Rahman, left Douma on Sunday, state media said.
On Saturday it concluded a week-long evacuation of thousands of its fighters from areas it had controlled in other parts of the enclave.
Opposition sources say Jaish al Islam officials have been desperately trying to strike a deal that would bring Russian military police into Douma, and let the group keep a role in maintaining internal security but under the state's overall jurisdiction.
Russia, a key ally of President Bashar al-Assad, has told the group it accepted such an arrangement but the Syrian government remained against it, a senior opposition source familiar with the talks said.
The Syrian army last week warned the insurgents to surrender or face a military assault to drive them out.
The Syrian army command said on Saturday it had regained most of the towns and villages in eastern Ghouta and was pressing its military operations in Douma.
The city's fall would seal the rebels' worst defeat since 2016.
The once bustling commercial hub on the outskirts of the capital was the main centre of street protests in the Damascus suburbs Assad's rule that ignited the conflict more than seven years ago.
Defence analysts say a major goal of the government's campaign was to complete a security belt around the capital, where for years rebels dug into an elaborate network of tunnels and well-fortified positions had resisted countless offensives to seize the enclave.
(This story has not been edited by NDTV staff and is auto-generated from a syndicated feed.)
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