State news agency SANA, citing police, said dry branches had caught fire and the blaze spread to a fuel tank which exploded.
"At least 27 people were killed ... And at least 30 people are being treated for burns," it said, citing a doctor at the scene.
It said the clinic had also suffered "significant material damage".
State television said most of the victims of the blaze at the Maysalun clinic in the majority Kurdish city of Qamishli were children.
Control of Qamishli is divided between Kurdish forces and regime troops loyal to President Bashar al-Assad.
The city has been relatively spared from the violence of Syria's conflict, which has left more than 220,000 people dead since it began in March 2011 with anti-government demonstrations.
