The advance by the government, which has been backed by a Russian air offensive, is seen as a major setback for opposition forces in Syria's second city.
The United Nations says about 600,000 Syrians already live under siege throughout the country, most encircled by government forces, though rebels and the Islamic State group have also used the tactic.
A US-backed alliance of Kurds and Arabs has also surrounded the IS stronghold of Manbij about 100 kilometres (60 miles) northeast of Aleppo city.
In Aleppo city itself, food and fuel shortages had already hit rebel-held eastern districts that are home to at least 200,000 people, after government forces advanced to within firing range of the key Castello Road supply route on July 7.
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Regime forces seized the road itself at the weekend, completely severing the rebel-held portion of the city from the outside world and raising fears of a potentially devastating siege.
"I don't know what is going to happen to us," said 38-year-old Mohamed Rukby, an unemployed father-of-four in the Bustan al-Qasr neighbourhood.
In the city's opposition-controlled Al-Mashhad district, mechanic Mohamed Zeitun said his work had dried up because a shortage of fuel meant residents were not driving anywhere.
"The idea of the siege keeps me up at night," the 44-year-old told AFP.
"I don't have supplies to last me more than a week, and if there is no food in the markets, there could be a famine," said the father-of-five.