Syria's ousted president Bashar al-Assad and his family are in Moscow, a Kremlin source told Russian news agencies, hours after he fled the country as Islamist-led rebels entered Damascus.

The announcement comes as Russia, a key Assad ally, called for an emergency meeting of the UN Security Council on the fast-changing situation on the ground in the war-torn country.

"Assad and members of his family have arrived in Moscow," the source told the TASS and Ria Novosti news agencies. "Russia granted them asylum on humanitarian grounds," he added.

Asked whether Assad was confirmed to be in Moscow, a Western official said they believed that was likely the case and had no reason to doubt Moscow's claim.

The Kremlin source also said the rebels who ousted Assad in a lightning offensive "guaranteed the security of Russian army bases and diplomatic institutions on Syria's territory".

Russia, Assad's biggest backer along with Iran, holds a naval base in Tartus and a military airfield in Khmeimim.

Moscow's forces became militarily involved in the Syrian conflict in 2015, providing support for Assad's forces to crush the opposition in the bloody civil war.

"Russia has always been in favour of a political solution to the Syrian crisis. Our starting point is the need to resume negotiations under the auspices of the UN," the Kremlin source added.

A Russian representative to the United Nations announced that Moscow had requested an emergency closed-door meeting of the UN Security Council on the situation in Syria for Monday afternoon.

"The consequences (of the events in Syria) for this country and the whole region have not yet been measured," the official said on Telegram.