Aid group Medecins Sans Frontieres said 28 other children were wounded in Saturday's attack on the school in the Sa’ada province.
The United Nations Children's Fund said the facility was a religious school in Juma'a Bin Fadil village in the province's Haydan district.
Dozens of airstrikes have hit civilian facilities in Yemen since a coalition of Arab states led by Saudi Arabia began military operations in March 2015 to restore fugitive former President Abd-Rabbu Mansour Hadi to power and roll back gains by Houthis.
A spokesman for Ban said he condemned the attack and urged parties "to prevent further violations of international humanitarian law and human rights and do everything in their power to protect civilians and civilian infrastructure."
A spokesman for the Saudi-led military coalition did not respond to requests by Reuters for a comment on the reports. But Saudi media quoted General Ahmed al-Assiri as claiming that the coalition had targeted a center used by the Houthi militias as a training camp.
The Saudi-led coalition intervened in Yemen after the Houthis advanced on Aden in 2015, forcing Hadi to flee the country.