The attacks on December 21 came just as preparations were being made for the arms at the sites to be destroyed outside Syria under a UN resolution decreeing the elimination of the country's chemical arsenal, the source said.
"On December 21, 2013, armed terrorist groups staged a major attack against one of the sites, located in central Syria... But the authorities fought back and the attack failed," said the source, using the regime's term "terrorists" to refer to rebels.
He also said rebels, among them Al-Nusra Front jihadists and the Salafist Jaysh al-Islam, attacked a second site in Damascus province.
They "tried to attack it using an armoured vehicle loaded with large amounts of explosives, but troops tasked with guarding the site fought them off and detonated the (vehicle) before it made it in."
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The attack, he added, killed four regulars and wounded another 28.
"Attempts (to attack) these sites are ongoing," said the source.
Meanwhile, he accused states that back the Syrian opposition of "leaking" information on the location of chemical weapons sites "and of encouraging the terrorists to attack" them.
"Syria holds these countries... Responsible for the danger... And for the catastrophic consequences" that such "leaks" could have, said the source.
The ministry source called on the UN and the Organisation for the Prohibition of Chemical Weapons (OPCW) to "make every possible effort for these irresponsible countries to stop their practices."
The OPCW has been working in Syria since October to eliminate its massive chemical stockpile after a UN Security Council resolution decreed the deadly arms must be destroyed.
The regime and its allies in Russia and Iran have systematically blamed rebels for the attacks, while the opposition and its supporters blamed Assad loyalists.
The UN concluded earlier this month that chemical weapons were used at least five times in the country's war before Syria agreed to dismantle its arsenal.