At least one person was reported killed in the fighting in Ikhlas neighborhood near the Council of Ministers, a huge complex, and a Damascus University campus, they said.
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Hundreds of families were fleeing the area, located between the districts of Kafar Souseh and Mezze, they said.
"The refugees have nowhere to go. There is fighting across Damascus," said a housewife watching the fighting from a tower block on Mezze Autostrade near the prime minister's office.
Fighting has been focused in the southern and north-eastern suburbs of the city, as well as the central areas of Mezze and Kafar Souseh where several security sites are located. Other parts of central Damascus were quiet on Thursday.
Another resident said army snipers were deployed on rooftops in Mezze and Kafar Souseh after rebels attacked armored vehicles stationed near the prime minister's office and a roadblock erected in the last few days behind the Iranian embassy.
"The snipers are shooting at anyone in the streets. Mezze streets are deserted," he said, speaking by phone.
Fighting was also reported in Midan, a central Sunni Muslim district where rebels have been operating in alleyways and narrow streets that cannot be entered easily by tanks.
Witnesses also said armoured vehicles entered the Sinaa neighbourhood, which is adjacent to the historic Old City centre of the ancient capital.
Opposition sources: Assad in Latakia
Assad is directing the response to the assassination of three of his top lieutenants from the coastal city of Latakia, opposition sources and a Western diplomat said on Thursday.
Assad, who has not made a public appearance since Wednesday's bombing, was commanding the government operation, they said. It was not clear whether Assad travelled to the Mediterranean sea resort before or after the attack.
"Our information is that he is at his palace in Latakia and that he may have been there for days," said a senior opposition figure, who declined to be named.
The palace, which Assad has used before to conduct official business, is located in hills near the city, Syria's main port.
The diplomat, who is following events in Syria, said: "Everyone is looking now at how well Assad can maintain the command structure. The killings yesterday were a huge blow, but not fatal."
Latakia province is home to several towns inhabited by members of Assad's minority Alawite sect.
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