The Syrian American Medical Society, which supports the hospital, said it had been struck by barrel bombs.
There are also reports of Russian-backed Syrian government forces hitting Aleppo's historic Old City.
Clashes between government troops and rebels are occurring in several areas.
Russian and Syrian air forces resumed attacks on the rebel-held east of the city after a partial truce lapsed on 19 September. Government forces have also launched a ground offensive against the rebels.