The Kremlin said on Monday President Vladimir Putin had made the decision to grant asylum in Russia to Bashar al-Assad, who was ousted as Syria's president by a lightning rebel offensive.

Assad's fall is a big blow to Iran and Russia, which had intervened in Syria's 13-year civil war to try to shore up his rule despite Western demands that he leave power.

Russian news agencies cited an unidentified Kremlin source on Sunday as saying Assad was in Moscow with his family.

"Such decisions cannot be made without the head of state. This is his decision," Kremlin spokesman Dmitry Peskov told reporters but gave no further details on Assad's movements.

Assad's departure removes a bastion from which Iran and Russia wielded power across the Middle East. Assad's father, Hafez al-Assad, sided with the Soviet Union to try to achieve parity with U.S.-backed Israel.

The Kremlin said Syria was subject to "extreme instability" and it was too early to speak about the future of Russian bases in the country.

"We are in dialogue with Ankara and other regional states, including on Syrian affairs," Peskov said.

"Indeed, Syria is going to have a very difficult period now, due to instability. And, of course, it is very important to maintain a dialogue with all regional countries here. We are determined to do this."

RUSSIAN BASES

It is unclear how Hayat Tahrir al-Sham, a former al-Qaeda affiliate that led the rebels' advance, will view Russia's military facilities in Syria - the Hmeimim airbase in Latakia province and a naval facility at Tartous on the coast.

"It's premature to talk about it yet," Peskov said. "This is all a subject for discussion with those who will be in power in Syria."

"Of course, everything is being done now that is necessary and everything that is possible in order to get in touch with those who can deal with security. And, of course, our military is also taking all necessary precautions," Peskov said.

The Tartous facility is Russia's only Mediterranean repair and replenishment hub and Moscow has used Syria as a staging post to fly military contractors in and out of Africa.

Moscow has supported Syria since the early days of the Cold War, recognising its independence in 1944 as Damascus sought to throw off French colonial rule. The West saw Syria as a Soviet satellite.

"We see the situation around Ukraine, we see many contradictory statements in this regard, we see a growing conflict potential in other regions, we can say the burning Middle East," Peskov said.