Syrian state television reported twin bombings near security service buildings in the centre of the capital on Sunday and said they wounded four people.
"A terrorist attack with two bombs occurred in al-Mehdi Street in the Abu Remmaneh district," the report said.
Several security service buildings are located in the area, which also houses the office of Vice-President Faruq al-Shara.
The attack took place near a security services building which protects the army's general staff, whose headquarters are located in the central Umayyad Square, some 3km away.
Abu Remmaneh is an upscale neighbourhood in the heart of Damascus, and is home to several embassies.
The Ahfad al-Rasul (Grandchildren of the Prophet) brigade of the rebel Free Syrian Army claimed responsibility for the attack in a statement on Facebook, in which it also threatened to attack President Bashar al-Assad's palace.
"This operation was carried out in response to the massacres in Daraya," said the statement, referring to the killing last week of at least 330 people in a town near Damascus.
Regime and rebel forces blamed each other for the massacre.
The group said it had prepared the twin bombings with the help of two other rebel groups.
Earlier on Sunday, state media reported that a car bomb explosion near a mosque at Sbeneh in the southern outskirts of the capital on Saturday killed 15 people.
Sbeneh is a poor neighbourhood where anti-government sentiment is strong.
The latest explosions come after a car blast in the southeastern suburb of Jaramana on August 28 which killed at least 27 people, according to the Syrian Observatory for Human Rights.