The departures began a month to the day after government forces launched a major offensive to retake all of Aleppo and secure the regime's biggest victory in more than five years of conflict.
Stating that there are still thousands of civilians and injured people in the besieged part of Aleppo, she shared images of people returning from the evacuation point.
The UN Security Council was to meet overnight to discuss the crisis in Aleppo in response to a request by France, which is calling for global observers to be sent to monitor the situation and ensure aid deliveries.
"We are engaged in very intense talks with the representatives of armed opposition, with Turkey also involved as a mediator", Putin said during a diplomatic visit to Japan.
The International Committee of the Red Cross, supervising the evacuations, said it was looking into reports of shooting before the convoy was turned around.
"Rebel sources accused pro-government Shi'ite militias of opening fire on bus convoys taking evacuees out of the city".
The Observatory said war planes bombed insurgent-controlled areas west of Aleppo overnight and north of Aleppo on Saturday.
"Now the people are afraid and they are running away from the meeting point where they were supposed to gather in order to take the buses out of the besieged east Aleppo districts, they are in a state of horror and shock", he said.
The Britain-based Syrian Observatory for Human Rights said pro-government fighters had blocked the road the evacuation convoy had been using and rebels were refusing the evacuation of residents of Fuaa and Kafraya.
Almost 15,000 people, including 4,000 rebels, were set to evacuate eastern Aleppo as part of a Russian-Turkish deal to end the presence of rebels in the city.
The second day of the operation to take fighters and civilians out of Aleppo's opposition enclave ground to a halt amid recriminations from all sides after a morning that had seen the pace of the operation pick up.
Syrian state TV put it at more than 9,000 while Russian Federation, a key Assad ally, said over 9,500 people, including more than 4,500 rebels, were taken out.
The Syrian government official, part of the evacuations negotiating team, said: "It was agreed to resume evacuations from east Aleppo in parallel with the evacuation of (medical) cases from Kefraya and al-Foua and some cases from Zabadani and Madaya".
Aleppo, a once-flourishing economic centre with renowned ancient sites, has been pulverised during the war that has killed more than 300,000 people, created the world's worst refugee crisis and allowed for the rise of Islamic State.
On Friday, Russian President Vladimir Putin said that he was working with Turkish President Recep Tayyip Erdogan to set up fresh peace negotiations. Their evacuation was a last-minute condition from Iran under the Aleppo agreement.