The Lebanese militant group's Al-Manar television said the raid hit a warehouse and fuel tanks, without specifying whether they were its own or belonged to the Syrian army or another of its allies.
Israel has carried out multiple air strikes in Syria since the civil war erupted in 2011, most of which it has said targeted arms convoys or warehouses of its Lebanese arch-foe Hezbollah.
In line with its usual practice, Israel's military declined to comment on the blast.
Al-Manar said preliminary reports suggested the blast caused only material damage and no casualties.
"Al-Manar's correspondent reported that an explosion struck at dawn on Thursday in fuel tanks and a warehouse near Damascus International Airport and that it was probably the result of an Israeli strike," the channel reported.
A resident of the Dawwar al-Baytara neighbourhood in the southeast of the capital, who lives in a tower block that looks towards the airport area, said he had seen an immense fireball.
"I ran to the balcony and looking towards the south,in the direction of the airport, I saw a huge fireball," he said.
"There was a power cut and it was pitch black, and the fireball was clearly visible."
The Israeli intelligence minister said the blast was in line with his government's policy of preventing advanced weaponry from reaching Hezbollah.
"We are acting to prevent the transfer of sophisticated weapons from Syria to Hezbollah in Lebanon by Iran," Katz told army radio.
"When we receive serious information about the intention to transfer weapons to Hezbollah, we will act. This incident is totally consistent with this policy."
The airport lies about 25 kilometres (15 miles) southeast of the city centre.
It was hit by Israeli air strikes in December 2014, Syrian state media reported at the time.
Israel does not usually confirm or deny each individual raid it carries out.
But last month, it said it had carried out several strikes near the Syrian desert city of Palmyra, targeting what it said were "advanced weapons" belonging to Hezbollah.
The strikes prompted Syria to launch ground to air missiles, one of which was intercepted over Israeli territory in the most serious flare-up between the two neighbours since the Syrian civil war began six years ago.
"When we identify attempts to transfer advanced weapons to Hezbollah and we have intelligence and it is operationally feasible, we act to prevent it," he said.
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