The brutal attack hit the bus in the city of Saada, whilst it was visiting a market.
A TV station run by the rebel Houthi movement reports that it was hit by a Saudi-led coalition air strike.
The International Committee of the Red Cross said medics in the Yemeni city of Saada, where the attack took place, had received the bodies of the children, all of them under 15 years of age.
Al-Malki insisted Thursday's attack carried out in Saada is a "legitimate military action" and is "in accordance with global humanitarian law and customs".
According to the spokesman, Guterres has stressed that "all parties must take constant care to spare civilians and civilian objects in the conduct of military operations".
"The targeting that happened today in Saada province was legal military action to target elements that planned and executed the targeting of civilians in the city of Jizan last night, killing and wounding civilians", the Saudi Press Agency cited al-Maliki as saying.
The Saudi-led coalition, meanwhile, said it targeted the rebels, known as Houthis, who had fired a missile at the kingdom's south on Wednesday, killing one person who was a Yemeni resident in the area.
The alliance also accused the Houthis of using children as human shields. Turki al-Malki said in a statement.More news: Trump attacks report that he’s anxious about son’s exposure to Mueller inquiry
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The war in the impoverished country has since left almost 10,000 people dead and unleashed what the United Nations describes as the world's worst humanitarian crisis.
Norway Refugee Council Secretary-General Jan Egeland described Thursday's airstrike differently: "grotesque, shameful, indignant".
In a statement, Save the Children condemned the attacks and called for a "full, immediate and independent investigation into this and other recent attacks on civilians and civilian infrastructure, including schools and hospitals". Houthi backed television stations broadcast footage of young children receiving treatment in local hospitals. Some shell-shocked boys are seen coming to, their faces bloodied and limbs charred.
UN Secretary-General Antonio Guterres urged an "independent and prompt" probe, while State Department spokeswoman Heather Nauert said the USA is "calling the Saudi led coalition to conduct a thorough and transparent investigation into the incident".
Yemeni tribal leaders told the AP the Saudi-led USA -backed coalition fighting Shiite Houthi rebels was behind the airstrike.
That was the toll at just one hospital.
"The victims were mostly pupils who were on their way to attend a summer school in Dahyan", he said.
"It is beyond cruel; innocent children's lives have been lost".