According to SANA, the shells slammed into an area around the Al-Mouwassat and Children's hospital in west Damascus, "leaving no casualties".
The hospitals are several hundred metres (yards) from Tishreen which is reserved for visiting dignitaries but is not an actual residence of President Bashar al-Assad.
This is the first time that the Syrian authorities have reported shells falling near a presidential palace.
The military council of the rebel Free Syrian Army meanwhile announced on Facebook that "the Free Army has fired mortars at the Tishreen presidential palace, resulting in a definite hit".
Tishreen is one of three such palaces in the capital. The others are the Peoples' palace atop Mount Qassioun in the north and Rawda palace in the centre, which holds the executive offices.