Four mortar shells struck the mainly Shia Shaghour neighborhood of the city, police told the Sana news agency.
Two of the shells hit the Badr al-Din al-Hussein technical institute.
The attack comes a day after President Bashar al-Assad registered to stand for re-election defying calls to step down as a way of ending Syria’s civil war.
Bashar al-Assad’s forces have pushed back rebels from many of their strongholds around the capital, but residents say they have responded by increasing the number of rocket and mortar attacks in the centre of the city.
“Twelve citizens were killed and 50 others wounded by terrorists who targeted the Shaghour neighborhood,” Sana reported.
The word “terrorists” is used by the Syrian authorities to describe all those seeking to depose President Bashar al-Assad’s government.
The Syrian Observatory for Human Rights has also reported the mortar attack, saying that the number of fatalities is expected to rise.
It says that the Badr al-Din al-Hussein technical institute is an Islamic law studies centre, with students as young as 14.
More than 150,000 people are believed to have been killed in the three-year civil war.
Millions of people have fled their homes as fighting shows no sign of easing. The UN says that almost 3.5 million civilians are being denied vital aid, including medicine and medical care.
A group of prominent lawyers and academics has urged the UN in an open letter to deliver aid into Syria with or without the government’s consent.
The letter, signed by 35 legal experts, says permission for aid is being arbitrarily withheld.
A UN resolution adopted in February called for all sides in the conflict to allow unrestricted humanitarian access.
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