The attack has been blamed on the Saudi-led coalition waging war against Yemen's Houthi rebels.
The US State Department called on Thursday for the Saudi-led coalition to investigate the air strikes.
The bus had been travelling through a busy market in Dahyan district, northern Saada, at the time of the raid, which Houthi rebels' Al Masirah TV reported killed 39 and wounded 51, "mostly children".
Earlier, Maliki accused the Houthis of "recruiting child soldiers, throwing them in battlefields and using them as tools".
Most of the images of the children hit by this airstrike are so gruesome the broadcasting rules don't allow us to show them - but a warning, the report still contains distressing images.
Most of the children were inside the bus when the airstrike hit, according to a medic, Yahya al-Hadi.
"Scores killed, even more injured, most under the age of ten", the head of the ICRC in Yemen, Johannes Bruwer, said on his Twitter account, adding that the ICRC in Yemen is "sending additional supplies to hospitals to cope with the influx".
In a statement Thursday, the Saudi-led coalition said the strike was a "legitimate military action to target elements that planned and carried out" an attack that targeted civilians in Jizan, a border city in southwest Saudi Arabia.
An official in Saada's medical office, who wished to remain anonymous, told Middle East Eye the air strikes killed 47 people with 77 more wounded, and that number was likely to rise.
"We are trying to get rid of the arms manifestations in the province, and to carry on the efforts exerted by the UAE to counter the terrorism groups in Yemen", Col Mohammed Salem Al Buhar, commander of the elite forces in Shabwa, told The National.
The United States and other Western powers provide arms and intelligence to the alliance, and human rights groups have criticized them over coalition air strikes that have killed hundreds of civilians at hospitals, schools and markets.
Turki al-Maliki said the kingdom's air defense had intercepted a ballistic missile fired by the Houthi rebels at a densely populated civilian area in Jizan, according to the Saudi Press Agency.
Saudi Arabia shot down the missile, with debris killing a Yemeni man and wounding 11 others, the coalition said.
"This is yet another example of the blatant violations of worldwide humanitarian law that we have seen in Yemen over the past three years-from indiscriminate and disproportionate attacks on civilians, denial of access to humanitarian aid and the use of starvation as a weapon of war-it's the people of Yemen, not the warring parties, who are paying the ultimate price".
After the strike, the United States, which largely supports the coalition's campaign, issued a statement.
Wednesday's attack brings the tally to 165 rebel missiles launched since 2015, according to the coalition, which that year joined the Yemeni government's fight against Huthi rebels.
In another video, which appears to show the immediate aftermath of the strike, several lifeless bodies of children lay under a blown-up bus.
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