The Syrian Observatory for Human Rights, a Britain-based war monitor, said Saturday's strikes hit a weapons depot outside the airport.
The news agency posted a video of a small explosion in the night sky on a backdrop of city lights, but provided no further details of the attack.
Its officials say they want a federal Syria that respects their autonomy from Damascus, the AP news agency reported.
An Israeli military spokeswoman declined to comment on the reported incident, when approached by Reuters, saying "we don't comment on foreign reports".
An alleged Israeli missile strike on the Damascus airport late Saturday reportedly targeted an Iranian plane delivering weapons for pro-regime forces and Revolutionary Guards units fighting in the country's civil war.
He had no immediate information on casualties.
Israel has said it will not tolerate Iran's presence in bordering Syria and has demanded its complete withdrawal.
Earlier this month, Israel's Defense Forces said it carried out more than 200 airstrikes in Syria over the previous year, many of which Israel claimed were targeting Iranian positions and weapons convoys.
Syria's conflict erupted in 2011 and has since killed more than 360,000 people, with millions more displaced internally and to neighbouring countries.
After losing swathes of territory to rebel groups, President Bashar Assad's troops have regained the upper hand and are now in control of around two-thirds of the ravaged country.
Syrian troops have amassed around the opposition bastion for weeks, although an expected assault appears to be on hold for now as regime ally Russian Federation and rebel backer Turkey try to hash out a deal on Idlib.