Russia's Foreign Ministry said earlier that Assad had left Syria and given orders for a peaceful transfer of power, after rebel fighters raced into Damascus unopposed on Sunday, ending nearly six decades of his family's iron-fisted rule.

"Syrian President Assad of Syria and members of his family have arrived in Moscow. Russia has granted them asylum on humanitarian grounds," the privately-owned Interfax news agency and state media quoted the unnamed Kremlin source as saying.

Interfax cited the same Kremlin source as saying Russia favoured a political solution to the crisis in Syria, where Moscow supported Assad during the long civil war.

The source said negotiations should be resumed under the auspices of the United Nations.

Syrian opposition leaders had agreed to guarantee the safety of Russian military bases and diplomatic institutions in Syria, the source told news agencies. However, some Russian war bloggers said the situation around the bases was extremely tense, and the source did not say how long the security guarantee lasted.

Moscow, a staunch backer of Assad, whom it intervened to help in 2015 in its biggest Middle East foray since the Soviet collapse, is scrambling to salvage its position. Its geopolitical clout in the wider region and two strategically-important military bases in Syria are on the line.

With AAP.