Some 2,000 people were being treated there, according to hospital officials, who believe there could be hundreds of casualties.
The Afghan health ministry’s spokesman, Sharafat Zaman Amarkhail, told the BBC there were no military facilities near the hospital.
Residents reported hearing loud explosions across Kabul at around 20:50 local time (16:20 GMT), followed by the sound of aircraft and air defence systems.
Family members of those being treated at the hospital were gathered outside, desperately trying to find information about their loved ones.
A Taliban government spokesperson claimed the death toll reached at least 400, but the BBC has not independently verified the numbers.
Pakistan’s information ministry said the strikes were “precise and carefully undertaken to ensure no collateral damage is inflicted”.
It dismissed Afghanistan’s claim as a “misreporting of facts… [that] seeks to stir sentiments, covering illegitimate support to cross-border terrorism”.