The reported raid on Tuesday hit an arms depot and wounded three soldiers, Syria's state-run news agency SANA quoted an unnamed military official as saying.
"Our air defences confronted hostile missiles launched by Israeli warplanes from above the Lebanese territories and downed most of them before reaching their targets," the military source said.
Lebanon's state-run National News Agency earlier reported that Israeli jets were flying at low altitude over parts of southern Lebanon.
Nearly an hour after the attacks began, Damascus residents could still hear the air defence units firing towards targets in the air.
There was no immediate comment by the Israeli military on the report.
"An IDF aerial defence system activated in response to an anti-aircraft missile launched from Syria," the official Israeli army Twitter account later said.
Israel has in the past carried out dozens of air raids in neighbouring Syria against what it says are Iranian targets, many of them south of Damascus.
At the end of November, Syria said its air defences had targeted and downed a number of "hostile targets" over the Kisweh area south of the capital.
In early December, state media reported that Syrian air defences had opened fire near Damascus airport but later withdrew the report without explanation.
Tuesday's reported attack near Damascus is the first since US President Donald Trump announced last week that Washington will withdraw all of its 2,000 forces in Syria.
Following Trump's surprise announcement, Israeli Prime Minister Benjamin Netanyahu has said that Israel would "continue to act against Iran's attempts to entrench itself militarily in Syria, and to the extent necessary, we will even expand our actions there".