The Russian Defense Ministry said militants could leave with their families and personal weapons through a secure corridor out of eastern Ghouta, where government forces are making rapid gains in a fierce assault.
"The Russian Reconciliation Centre guarantees the immunity of all rebel fighters who take the decision to leave eastern Ghouta with personal weapons and together with their families,” said the Defense Ministry statement, according to Reuters.
Vehicles would "be provided, and the entire route will be guarded", it added.
The Syrian army has captured more than a third of the enclave in recent days. It has pressed ahead as Takfiri militants continue to shell Damascus.
Syrian President Bashar Assad said on Sunday the Syrian army would continue the push into eastern Ghouta, an area of farmland and towns just outside Damascus which government forces have encircled since 2013.
In the past two weeks, Eastern Ghouta has witnessed renewed violence with terrorists mounting repeated mortar attacks on the Syrian capital in the face of an imminent rout. Western powers, however, blame the Syrian government and Russia for the crisis.
The Syrian air force recently dropped millions of leaflets on the district informing civilians about a safe humanitarian corridor prepared for their evacuation under a Russian-proposed humanitarian pause.
The leaflets contained instructions for the citizens urging them to leave Eastern Ghouta through al-Wafideen Camp.
Takfiri militants are under siege in Eastern Ghouta, along with an estimated 400,000 civilians.
On Sunday, The Russian military said militants based in the Damascus suburb have imposed a curfew during humanitarian pauses to prevent civilians from leaving the enclave.
The militants have also banned mass gatherings of civilians, Interfax quoted Russian Major-General Vladimir Zolotukhin as saying.
Russia has designated four safe passage routes in Eastern Ghouta after a ceasefire was declared across Syria by the United Nations Security Council.