Lt Gen Zaid al-Saleh, head of the government's local security committee, said rebel fighters did not "have much time" and needed to "surrender or die".
Tens of thousands of civilians are also believed to be in the rebel enclave.
Syrian state TV showed footage of people in Aleppo celebrating, amid reports the army was close to victory.
The director of the UK-based monitoring group, the Syrian Observatory for Human Rights, warned of "real massacres" being carried out in Aleppo.
Rami Abdel Rahman called on the international community to find a safe haven for civilians.
Rebels have now lost more than 90% of the territory they once held in eastern Aleppo since government forces stepped up their offensive to regain full control of the city a month ago.
Russia, which backs the government, says more than 100,000 civilians have been displaced by the fighting and that 2,200 rebel fighters have surrendered.
Aleppo was once Syria's largest city, and its commercial and industrial hub before the uprising against President Bashar al-Assad began in 2011.
For much of the past four years it has been divided roughly in two, with the government controlling the western half and rebels the east.
Troops finally broke the deadlock with the help of Iranian-backed militias and Russian air strikes, reinstating a siege on the east in early September and launching an all-out assault weeks later.
On Monday morning, the official Sana news agency cited a military source as saying that the army had taken full control of the key southern district of Sheikh Saeed, as well as the neighbouring areas of Karam al-Daadaa and Saliheen.
Hours later, nearby Bustan al-Qasr, Kallasa, Fardous, Jaloun and Jisr al-Haj had also fallen after rebel fighters withdrew in the face of an intense government bombardment, the Syrian Observatory for Human Rights reported.
Late on Monday the monitoring group said that clashes were continuing in Salah al-Din neighbourhood and other areas remaining under rebel control.
In an interview with the BBC's Victoria Derbyshire programme, an English teacher who is still inside a rebel-held area described the conditions as terrible.
"The situation inside the eastern part of Aleppo is literally doomsday," Abdul Kafi Alhamado said. "Bombs are everywhere, people are running, people are injured in the streets, no-one can dare go to help them, some people are under the rubble."
The Syrian Observatory says that at least 415 civilians and 364 rebel fighters have been killed in rebel-held areas since 15 November. Another 130 civilians have died in rebel rocket and mortar attacks on the government-controlled west.
Russia and the United States, which backs the rebels, held talks in Geneva over the weekend to discuss a deal for civilians and rebel fighters to leave Aleppo.
But on Monday, US officials said their Russian counterparts had rejected a proposal for an immediate cessation of hostilities to allow for safe departures.
Analysts say the fall of Aleppo would be a big blow to the opposition, as it would leave the government in control of Syria's four largest cities.
However, the head of the umbrella group that represented political and armed opposition factions at failed peace talks at the start of this year insisted that their determination to overthrow the president would not be diminished.
"If Assad and his allies think that a military advance in certain quarters of Aleppo will signify that we will make concessions, then [I say] that will not happen," Riyad Hijab, general co-ordinator of the High Negotiations Committee, told reporters.
As fighting intensified for the last rebel-held neighbourhoods, there have been frantic efforts to secure guarantees to evacuate the most vulnerable - the very ill, gravely injured and innocent children. All efforts have been in vain.
"It's a far dream," one frustrated aid official told me on Monday, about a long-standing request for a UN-organised medical mission for about 500 patients and their families.
Two efforts, which seemed close to agreement, collapsed last week when Russia asserted there was no need for a ceasefire since tens of thousands were escaping without one.  Sources say Russia then shifted its stance. But rebel groups demanded a ceasefire and a delivery of aid first.
In desperation, aid agencies reduced their request from 72 hours to just three. A group known as "Doctors Under Fire" has also been pleading for the evacuation of hundreds of children.
British surgeon Dr David Nott says "we only need a temporary cessation of 60 minutes to save them all." Even that seems to have been too much.
Syria's army says the battle for Aleppo is in its final phase, after major gains by troops in the city's south left rebels on the brink of defeat.