The Turkish foreign minister said 37,500 people had already left the Syrian city and evacuations could be completed by Wednesday.
Turkey's foreign minister on Tuesday said 37,500 people have been evacuated from the Syrian city of Aleppo so far.
The plan was to complete all evacuations by Wednesday, Mevlut Cavusoglu said in a tweet from Moscow where he was holding talks with his Russian and Iranian counterparts.
Russian Foreign Minister Sergei Lavrov also said that the evacuation should finish within a couple of days.
"Right now the evacuation is wrapping up," Lavrov said.
Thousands of civilians are still waiting in dire conditions to be transported to the relative safety of the west of the city.
The evacuations began again after rebels allowed civilians to leave two besieged pro-regime Shia villages of Foua and Kefraya in nearby Idlib province, following their suspension over the weekend.
The UN on Tuesday said some 750 people had left those two villages as 20 buses headed there to continue to ferry out people.
The Syrian regime said it has authorised the United Nations to send an additional 20 staff to eastern Aleppo to monitor the ongoing evacuation, a UN spokesman said on Tuesday.
The UN Security Council on Monday unanimously called for UN officials and others to observe the evacuation of people from the last rebel-held enclave in Aleppo and monitor the safety of civilians who remain in the Syrian city.
As the evacuations proceed, the foreign and defence ministers of Russia and Iran – two key supporters of the Assad regime – and Turkey are meeting in Moscow to discuss the future of Syria.
The talks, aimed at giving a fresh start to seeking a solution to the slaughter in Aleppo, went ahead despite the assassination of Russia's ambassador to Ankara on Monday.
"There are still thousands. It's a huge crowd which includes women and children," International Committee of the Red Cross spokesman Krista Armstrong said of Aleppo.
Families in Aleppo have spent hours in below-freezing temperatures, sheltering from the rain in bombed-out apartment blocks desperately waiting for news of a new wave of departures.
Those evacuated so far included seven-year-old Bana al-Abed, whose Twitter account offered a tragic account of Syria's nearly six-year war, as well as 47 children who had been trapped in an orphanage.
The evacuation of civilians from the two villages had been demanded by the Syrian army and its allies before they would allow fighters and civilians trapped in Aleppo to depart.
The stand-off halted the Aleppo evacuation over the weekend.
Complex evacuations from East Aleppo and Foua & Kefraya now in full swing. More than 900 buses needed to evacuate all. We must not fail.— Jan Egeland (@NRC_Egeland) December 19, 2016
After an agonising delay, the operation resumed under a complex agreement that will see regime forces exert full control over Syria's second city.
TRT World's Adiz Tiyansan reports from western Aleppo.