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.
Israeli Intelligence Minister Yisrael Katz said it was consistent with his government's policy to prevent arms transfers to Hezbollah, but stopped short of confirming his country was behind it.
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.
"Around 4:00 am (0100 GMT), I heard an enormous explosion," Maytham, 47, told AFP.
"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."
- 'Consistent' with Israeli policy -
The Israeli intelligence minister said the blast was in line with his government's policy of preventing advanced weaponry from reaching Hezbollah.
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"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."
Israeli warplanes have hit the airport and other bases around the capital in the past, targeting what it said were weapons stockpiles destined for Hezbollah.
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.
Israeli Prime Minister Benjamin Netanyahu had confirmed those strikes and vowed there would more of necessary.
"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.
"That's how we shall continue to act," he added.
On January 13, Syria accused Israel of bombing Mazzeh airbase in the western suburbs of the capital. There were several strikes near the same base last year.
In April 2016, Netanyahu admitted for the first time that Israel had attacked dozens of convoys in Syria that he said were transporting weapons destined for Hezbollah.
Israel and Syria are still technically at war, though the armistice line had remained largely quiet for decades until 2011 when the Syrian conflict began.
Israel and Hezbollah fought a 34-day war in 2006 which killed 1,200 people in Lebanon, mainly civilians, and 160 Israelis, mostly soldiers.