"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," said the source, who is part of the evacuation's negotiating team.
The villages of al-Foua and Kefraya in Idlib province are besieged by insurgents. The towns of Madaya and Zabadani are blockaded by pro-government forces.
The opposition's Britain-based Syrian Observatory for Human Rights said the evacuation of some 4,000 people, including wounded, from the villages of Foua and Kfarya are to start Saturday.
Hezbollah fighters have joined the Syrian war fighting along with Syrian President Bashar al-Assad's forces. Opposition activists blamed the Lebanese group for blocking the main road south of Aleppo and blocking evacuations from rebel-held eastern neighborhoods of the city.
The Aleppo evacuation was suspended Friday after a report of shooting at a crossing point into the enclave by both sides of the conflict. Thousands were evacuated before the process was suspended. The Syrian government said the village evacuations and the one in eastern Aleppo must be done simultaneously, but the rebels say there's no connection.
Hezbollah's Military Media said the new deal includes the rebel-held towns of Madaya and Zabadani near the border with Lebanon, where tens of thousands of people are trapped under siege by government forces and the Lebanese group.
Syrian State TV correspondent, speaking from Aleppo, said Saturday that the main condition for the Aleppo evacuation to resume is for residents of Foua and Kfarya to be allowed to leave.
The cease-fire and evacuation from east Aleppo earlier this week marked the end of the rebels' most important stronghold in the 5-year-old civil war. The suspension demonstrated the fragility of the cease-fire deal, in which civilians and fighters in the few remaining blocks of the rebel enclave were to be taken to opposition-held territory nearby.
Syrian state TV said Friday that that the evacuation's suspension was due to rebels trying to smuggle out captives who had been seized in the enclave after ferocious battles with troops supporting Assad.
Reports differed on how many people remain in the Aleppo enclave, ranging from 15,000 to 40,000 civilians, along with an estimated 6,000 fighters.
There also were contradictory reports on the number of evacuees who left on Thursday and early Friday from east Aleppo. Syrian state TV put it at more than 9,000 while Russia, a key Assad ally, said over 9,500 people, including more than 4,500 rebels, were taken out.
The Red Cross wants safety guarantees for evacuees
The International Committee of the Red Cross (ICRC) has also intervened in the evacuation, asking Syria's warring parties to agree quickly on a plan to evacuate people from east Aleppo and provide "solid" safety guarantees.
"We're ready to resume facilitating the evacuation according to our humanitarian mandate. But we now expect all the parties on the ground to provide us with solid guarantees in order to keep the operation going. They're the ones who have to protect the people and provide safe passage," said ICRC head in Syria Marianne Gasser in a statement from Aleppo.
The ICRC said thousands of cold and scared people including women, children, the sick and injured, remain in eastern Aleppo waiting for the evacuation operation to continue.
"People are expecting us to continue the evacuation. It's important that the parties on the ground do their utmost to end this limbo," Gasser said. "People have suffered a lot. Please come to an agreement and help save thousands of lives."
The ICRC said it had managed to evacuate around 10,000 people, many of whom were in a critical medical condition.
The World Health Organization (WHO) representative in Syria Elizabeth Hoff, speaking from west Aleppo, told Reuters she was not sure there was time for the evacuations to begin on Saturday.
"It is a quarter past one in the afternoon already. The Russians (who are overseeing the operation) are not keen on working through the night," Hoff said.
The WHO and other aid workers are not yet present at the transit point in east Aleppo, Hoff said