The announcement comes more than a week into a regime assault to oust Islamic State group fighters from the capital's southern suburbs, including the Palestinian refugee camp of Yarmuk.
On Sunday, state news agency SANA said a deal had been reached to evacuate opposition fighters and members of their families from rebel-held areas east of Yarmuk.
SANA reported "an agreement reached between the Syrian government and terrorist groups in southern Damascus, in the areas of Yalda, Babila and Beit Sahem", using its usual term for all rebels.
The deal gives fighters the choice between leaving the area with their families or handing over their weapons and staying, SANA said.
The reported deal is the latest in a string of such agreements that have seen the regime retake areas near the capital after rebel withdrawals.
Such a deal around Yalda could allow the regime to deploy forces on the eastern edges of Yarmuk after other units advanced towards the camp from the west, a Britain-based war monitor said.
Over the past two days, the army has retaken large parts of the district of Qadam on Yarmuk's western flank, the Syrian Observatory for Human Rights added.
On Saturday, IS seized a hospital and surrounding buildings on the eastern edges of Yarmuk as it tried to push towards Yalda, the Observatory said.
On Sunday, regime war planes pounded Yarmuk and the neighbouring district of Hajar al-Aswad, it said.
Yarmuk and its surroundings are now the jihadist group's largest urban redoubt in Syria and neighbouring Iraq, after IS lost most of the swathes of territory it once held in both countries.
The jihadists have held parts of Yarmuk and Hajar al-Aswad since 2015, and overran Qadam in a surprise assault last month.
At least 85 regime fighters and 74 IS jihadists have been killed in ten days of fighting in southern Damascus, the Observatory says.
The announcement of an evacuation deal on Yalda and nearby areas comes after the regime reconquered what was once a key rebel bastion east of Damascus earlier this month.
Eastern Ghouta fell after a brutal military operation and a series of similar evacuation deals brokered by regime ally Russia that saw tens of thousands of residents bussed to northern Syria.
More than 350,000 people have been killed and millions displaced since the start of Syria's civil war in 2011 with a deadly crackdown on anti-government protests.
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