Media reports said the explosions were caused by car bombs and happened in the Kfar Sousa district of the capital close to a complex housing security and intelligence services.
Unconfirmed reports are that among the dead were detainees who were killed in buses near the scene, along with military personnel. Members of the Arab League delegation which is in the country to assess the situation, also visited the scene.
Sameer Seif el-Yazal, head of the first observers party, said: "We are here to see the facts on the ground... What we are seeing today is regretful, the important thing is for things to calm down."
Residents in Damascus reported hearing gunfire and ambulance sirens for a few minutes following the explosions.
State TV suggested the blasts could be the work of al Qaeda-linked militants. The Syrian state has blamed terrorists and foreign interference for the ongoing protests against President Bashar al Assad. Read More