In gunfire that followed the explosions, a television correspondent Maya Nasser of Press TV, the Iranian English-language satellite network was killed during a live broadcast, the network said. Press TV said its Damascus bureau chief, Hussein Murtada, also came under attack and was injured.
The explosions struck a warren of government buildings in one of the capital's most guarded areas, near a presidential office used by Mr. Assad. One of the bombs went off at a building used by the Army General Command, and the second hit the air force's command headquarters, witnesses said.
A huge pall of dark smoke rose over central Damascus afterward and was called the biggest since the explosions on July 18 that killed several of Mr. Assad's key security aides, including the defense minister and the president's brother-in-law.
There was no immediate claim of responsibility for the attack Wednesday.
The latest bombing attacks in and around Damascus on Tuesday and Wednesday seemed to demonstrate that antigovernment fighters remain able to strike close to centers of power, even after weeks of efforts by the Syrian military to drive the rebels from the capital and its suburbs.
On Tuesday, insurgents bombed a school they claimed was being used as a headquarters and barracks for military officers and for the government's plainclothes security enforcers, known as shabiha.
In a statement after the bombing on Wednesday, the Syrian Army said that "armed terrorist gangs, with foreign links, carried out a new terrorist act this morning by blowing up an explosives-laden car and a bomb at the Army General Command, which damaged the building, caused a fire and wounded some of the guards."
It was impossible to immediately verify how many people were hurt or killed in the explosions. Witnesses said they saw dead guards, soldiers and insurgent fighters. The Associated Press reported four dead and 14 wounded, citing state television, but the state news agency initially said only that guards at the military complex were injured.
Hala Jaber, a journalist in Damascus for The Sunday Times of London, posted photographs online Wednesday showing a fire at the military headquarters and smoke engulfing the facility. "Huge explosion now followed by sound of gunfire and siren of ambulances," Ms. Jaber wrote on Twitter, saying that a second explosion was heard shortly afterward.
Witnesses near the site of the bombings said the explosions were precursors to a fierce armed attack. One witness said that dozens of insurgent fighters appeared almost immediately after the blasts, attacking with grenades and gunfire while guards from the military complex ran around in a panic.
A 40-year-old driver who works near Ummayad Square, where the buildings are, described chaotic scenes of battle, with some rebels fighters fighting their way into the General Staff headquarters while other insurgents fired on soldiers from rooftops. Then, he said, "big numbers of soldiers and security men came from everywhere and they surrounded the building, and began fighting in a crazy way."
"I cannot believe what I saw," said the driver. "The government cannot protect its military and key security buildings, so how can it protect the country?"
Another witness described the shock of seeing Syria's conflict in the center of the capital, where it has rarely intruded. "I saw dozens of armed security men and soldiers killed and injured," he said, adding that, for a time, gunfire from the rebels prevented ambulances from approaching.
"To see dozens of armed rebels, with snipers, machine guns and grenades I was not expecting this in my life," the witness said.
The government responded quickly, shutting down parts of the capital, including entrances to Damascus and all roads leading to Ummayad Square. Temporary checkpoints were set up throughout the city center as dozens of security officers streamed in. The shabiha militiamen searched shoppers in the Souk al-Hamadiya, the market in the heart of old Damascus.