A rebel commander, who gave his name only as Khaldoun, said snipers from his brigade hit the wheels of Syrian Airways flight RB201 on Thursday.
"Those were warning shots," he said, adding that the plane had been unable to take off.
"We wanted to send a message to the regime that all their planes – military and civilian – are within our reach."
There were no immediate reports of the incident on Syrian state media.
Rebels accuse the government of using civilian aircraft to transport weapons and Iranian fighters they claim are helping President Bashar al-Assad's forces.
Insurgents have cut off many of the road links to Aleppo, Syria's biggest city.
Fighting around Damascus has made the road to the capital's international airport unsafe for traffic and foreign airlines have stopped flying there.
According to flight schedules, the Cairo-bound RB201 usually flies from Damascus rather than Aleppo.
Another rebel leader has urged civilians not to use Aleppo airport or Syrian Air flights "as they will be targets from now on".