A spokesman for the Afghan interior ministry told the Associated Press news agency the school in Sarobi was hit by a mortar shell and blamed Taleban supporters for the incident.
But he admitted there was no evidence to back up the claim and there has been no independent confirmation of the cause of the blast.
A BBC correspondent in Kabul says there could be other explanations - there is much unexploded ordnance lying around.
Rossella Miccio, an administrator at the Italian-run hospital where 18 of the injured were taken, said children as young as eight years old were hurt.
'Sabotaging stability'
"The injuries were mostly to (the children's) legs and abdominal parts," said one man, who travelled the 50 kilometres (30 miles) to Kabul to take three injured nephews for treatment.
The interior ministry spokesman blamed Taleban and al-Qaeda rebels who "want to sabotage the security and stability of this government of Hamid Karzai".
Correspondents say the area around Sarobi has often been a base for armed renegade groups.
It was there that four Western journalists were robbed and murdered last November while they were covering the American-led campaign against the Taleban.