Iran's state-supported Press TV said one of its correspondents was killed by gunfire while reporting and a second correspondent was injured, The New York Times said.
Iran is the chief regional ally of Syrian President Bashar Assad.
Syrian officials said an operation was under way to clear the building of terrorists, and that military officials were fine, CNN reported. The building houses the offices for the Joint Chiefs of Staff.
Witnesses said at least two large explosions rattled the military headquarters in central Damascus, setting fire to the building, the Times said.
The Free Syrian Army, a key group battling Assad's forces, took responsibility for Wednesday's attack, CNN said.
On Tuesday, rebels attacked a Damascus school building used by Syrian intelligence officials.
The government blamed both assaults on "terrorists," a term it often uses to describe anti-Assad fighters.
Syria's state-run news agency said bombs exploded in the school building Tuesday, wounding seven people and damaging the school.
The Grandchildren of the Prophet rebel group claimed responsibility for the attack, saying it planted nine bombs and killed senior leaders and security agents, however, that information couldn't be independently verified, the Times said.
Fighting was reported between government forces and rebels near Syria's border with Israel. Military officers reported no injuries or damage from mortar shells landing near a kibbutz in the Golan Heights.
Israeli officials said they were concerned about the expanding battlefield, the Times said.
"The shells were aimed at villages inside Syria and are actually part of the internal, ongoing conflict inside Syria," the officials said in a statement. "Fire from Syria leaking into Israel will not be accepted."