The Saudi-led coalition, meanwhile, said it targeted the rebels, known as Houthis, who had fired a missile at the kingdom's south the previous day, killing one person.
Yemen's rebel-run TV channel aired dramatic images of wounded children, their clothes and schoolbags covered with blood as they lay on hospital stretchers.
According to Yemeni elders, the attack took place in the Dahyan market in Saada province, a Houthi stronghold. The province lies along the border with Saudi Arabia. The bus was ferrying local civilians, including many school children, they said, speaking on condition of anonymity for fear of reprisals.
There was no breakdown in the casualties and it was not immediately clear how many of the victims were on the bus itself and how many were pedestrians in the immediate area around it. It was also unclear if there were other airstrikes in the area.
Col. Turki al-Malki, a spokesperson for the Saudi-led coalition, said the attack in Saada targeted the rebels who had fired a missile at the kingdom's south, killing one person and wounding 11 others. The coalition said Wednesday's projectile, fired toward the southwestern Saudi city of Jizan, was intercepted and destroyed but its fragments caused the casualties.
The statement, carried by the official Saudi Press Agency, also said the missile was launched "deliberately to target residential and populated areas." Al-Malki insisted Thursday's attack in Saada was carried out "in accordance with international and humanitarian law and customs."
Saudi Arabia backs Yemen's internationally recognized government and has been at war with the Houthis since March 2015. The rebels control much of northern Yemen, including the capital, Sanaa.