State television quoted the army as saying several rockets landed in the airport compound in Mezzeh, Reuters news agency reports, the BBC wrote.
It remains unclear if the alleged bombardment, which caused fires, has caused any casualties.
The Israeli government has previously neither confirmed nor denied that it carries out strikes in Syria.
It is thought to have bombed weapons shipments intended for Lebanon's Hezbollah movement several times since Syria's civil war began in 2011.
Read alsoIsraeli satellite spots Russian Iskander missiles - mediaHezbollah, which fought a month-long war with Israel in 2006, has sent thousands of fighters to Syria to support President Bashar al-Assad's forces.
State-run Syrian news agency Sana reported explosions at the Mezzeh military airport and said ambulances were rushing to the scene.
The army was quoted as warning Israel "of the repercussions of the flagrant attack", according to Reuters.
In December, Syrian state media said an Israeli missile strike had targeted the airport in Mezzeh, causing fires but no casualties.
Days earlier, on November 30, Israeli jets were reported to have fired missiles from Lebanese airspace into Syria, striking Sabboura, an area outside Damascus.
The target was unclear but the highway from Lebanon to Damascus runs through the town.
