Syrian state TV said it expected the evacuation of thousands of civilians and fighters from the last opposition footholds in Aleppo to be completed by tomorrow.
The departure of the last rebels from Aleppo would close another chapter in Syria's civil war and would give President Bashar Assad a significant symbolic and strategic victory.
Almost six years after the outbreak of an armed rebellion against Assad, the Syrian leader will be in charge again of the country's five largest cities and the Mediterranean coast.
The leaders noted that a quick launch of talks between the Syrian government and the opposition in Kazakhstan's capital, Astana, would be an important step toward that goal, a Kremlin statement said.
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The conversation came a day before a scheduled meeting of foreign and defense ministers of Russia, Turkey and Iran in Moscow. Russia and Iran have backed Assad, while Turkey has supported the opposition.
At the United Nations, officials said more than 100 UN humanitarian staff on the ground in Syria, most of them Syrian nationals, could start monitoring the Aleppo evacuations.
Earlier today, the UN Security Council had approved a resolution urging the immediate deployment of the monitors, following a French-Russian compromise over the text.
UN spokesman Stephane Dujarric said there will also be observers from the International Committee of the Red Cross and the Syrian Arab Crescent.
The resolution also demands that all parties allow unconditional and immediate access for the UN and its partners to deliver humanitarian aid and medical care, and "respect and protect all civilians across Aleppo and throughout Syria."
Syria's UN Ambassador, Bashar al-Ja'afari, claimed that one of the "main purposes" of the deployment was to rescue foreign intelligence officers from the former rebel-held area.