The bombs went off in the city of Homs, in a neighbourhood dominated by minority Alawities, a Shiite offshoot sect that President Bashar Assad also belongs to. It was one of the deadliest strikes to hit the government-controlled area in months.
The blasts occurred just as the children were leaving at the end of class at the Ekremah al-Makhzoumi elementary school, according to an official with the Homs governorate.
The explosions also wounded at least 56 people, said the local official who spoke on condition of anonymity because he was not authorised to speak to the media.
The first explosion was from a car bomb parked and detonated in front of the school, followed minutes later by a suicide bomber who drive by and detonated his explosives-laden car, said the official.
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The Britain-based Syrian Observatory also reported the bombings but said 18 people were killed and 40 wounded, though it did not say how many children were among the casualties. Conflicting tolls are common after bombings.
Homs Governor Talal Barazzi described the blasts as a "terrorist act and a desperate attempt that targeted school children."
There was no immediate claim of responsibility for today's attack but Syrian rebels fighting to oust Assad have carried out such bombings during the country's civil war.
There have been horrific attacks against civilians by all sides throughout the brutal conflict, now in its fourth year, but rarely have children appeared to be the direct target.
In May, Syrian government forces dropped a bomb in the northern city of Aleppo, hitting a complex that held a school alongside a rebel compound. At least 19 people, including 10 children were killed in that incident.