The Syrian Observatory for Human Rights said it was a "limited advance" but the first by government troops in more than a year, and that residents fearing a major operation were fleeing the region.
The troops have gained some ground in Aleppo in the past few weeks, taking advantage of the fact that rebels who hold larges swathes of territory have turned their guns against jihadist fighters.
"The Syrian army made a limited advance, for the first time in more than a year, and seized the Karm al-Qasr district on the southeastern fringes of Aleppo," held by the rebels, said the Observatory.
Al-Watan newspaper, which is close to President Bashar al-Assad's government, said the troops made the advance on yesterday and also seized the districts of Ballura and Kasr al-Tarrab.
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It too said the operation had been launched from Nairab airport in the east, as well as Aziza village in the south, while adding it had reached the outskirts of Mayssar, a rebel bastion in southeast Aleppo.
The Observatory said residents of Mayssar, nearby Marjeh and Enzarrat were fleeing their homes for "neighbourhoods controlled by regime forces... Because of the fighting".
Since December 15, Aleppo has been the target of an aerial offensive that has killed hundreds, mostly civilians.
Today at least three people were killed in regime air raids that targeted Qadi Askar and Mayssar, said the Observatory.
On Saturday, Aleppo-based activist Mohammad Wissam said the army has taken advantage of the fact that rebels have recently turned their sights against jihadists.
"The army has deployed en masse around the east of the city, and families residing in the area have started to flee for other safer neighbourhoods and villages" fearing an all-out assault, he said.
Control of the city is now divided between the regime and rebels, and the opposition also holds much of Aleppo province.
In other violence a Saudi suicide bomber and fighter from the Al-Nusra Front jihadist group blew himself up at an army checkpoint late yesterday, killing 13 soldiers and pro-regime militiamen, the Observatory said.