The first of the bombings in Damascus was near a mosque in the north of the city. It killed five security personnel and one civilian, state television said.
A motorcycle bomb struck as worshippers left a mosque after weekly prayers in the northern neighbourhood of Rokn Eddin.
“The terrorist attack killed five members of the security forces and injured several others,” state media said.
The second bomb went off several hours later near the Ministries of Justice and Information.
The Britain Syrian Observatory for Human Rights said another blast struck the Salhiyeh district.
No casualties were reported as a result of these blasts.
Witnesses said the capital’s southern suburb of Tadamun and neighbouring Palestinian refugee camp of Yarmuk came under heavy shellfire.
On the south-east outskirts of Damascus, hundreds of troops backed by armour stormed the town of Babila, where Free Syrian Army rebels were entrenched, the observatory said.
There have also been reports of more violence in Aleppo as government forces and rebels continue to battle for control.
In Syria’s commercial capital Aleppo in the north, one rebel was killed in clashes with the army in the embattled Salaheddin district, and fighting was also reported in nearby Izaa, the observatory said.
It also reported heavy shelling of rebel-held areas in the Salaheddin and Tarik al-Bab districts.
A journalist with the news network CNN, Nick Paton Walsh, says in the four or five days he spent there, government shelling appeared to be shelling residential areas.
“The aftermath of these air strikes [we're] watching civilians pulling out a lot of the time very, very, very young victims,” he said.
The conflict in Syria has claimed a total of more than 26,000 lives since it erupted in mid-March 2011, according to Observatory figures, with civilians accounting for most of those killed.
Brussels announced an extra $63 million for civilians trapped in the conflict as the European Union foreign ministers opened a two-day meeting in the Cypriot resort of Paphos, barely 100 kilometres from Syria.
At the meeting ministers called for a massive boost in aid to Syrian civilians amid mounting fears the humanitarian crisis could impact Europe.
Source: ABC News