The insurgents, who have made dramatic gains over the last week, are said to have entered the country's third-largest city.
Sources told the Reuters news agency that large army convoys have retreated from the city center, with senior commanders escaping the onslaught in helicopters.
As other rebel groups in the south-west advanced to within 30 km of the capital, the Syrian army said it was reinforcing its defenses in Damascus and further to the south.
State media reported that President Bashar al-Assad is still in the capital, but sources told American broadcaster CNN that he was “nowhere to be found”.
British newspaper the Telegraph reported that the Syrian leader’s family has fled to Russia.
Trump: U.S. should 'not get involved'
The civil war in Syria started in 2011 following an uprising against Assad's regime.
Over the last few years, the president - who has had the military backing of Russia and Iran - has been in charge of most of the country, although rebel groups remained strong in many areas, especially in the north.
One of those groups, the Islamist Hayat Tahrir al-Sham (HTS), is driving the latest assault, seizing the cities of Aleppo and Hama in a matter of days.
U.S. president-elect Donald Trump said on his social media platform that his country - which has supported Kurdish-led rebels in Syria's north - should avoid intervening in the conflict.
"Syria is a mess, but is not our friend, & THE UNITED STATES SHOULD HAVE NOTHING TO DO WITH IT. THIS IS NOT OUR FIGHT. LET IT PLAY OUT. DO NOT GET INVOLVED!," he wrote on Truth Social.
Earlier this week, Assad's Russian allies conducted air strikes in Syria for the first time in years, in response to the latest insurgency.
However, it is not clear whether Moscow could intensify its campaign to support Assad's forces, according to the Reuters news agency.
Russia has a significant airbase and naval base in Syria, but since 2022 its military has been tied up by its full-scale invasion of Ukraine.
Sources close to the Kremlin reportedly told the Telegraph that Russia is unwilling to step in to rescue Assad’s 24-year rule.
Iran and its proxies, meanwhile, are involved in conflicts with Israel in Lebanon and the Gaza Strip.
'A new reality'
Eyewitness reports from several parts of Syria suggest that support for Assad’s regime may be fraying.
Damascus residents told Reuters that protesters ripped up posters of the president, with some of them accompanied by soldiers who had deserted.
There were reports of desertions in the south also. In the east, around 2,000 Syrian soldiers crossed to border into the Iraqi town of al-Qa'im in order to seek sanctuary, according to the local mayor.
The foreign minister of Turkey - which backs some of the rebels - met his Russian and Iranian counterparts on Saturday and agreed on the importance of Syria's territorial integrity and on restarting a political process, Reuters reported.
Turkish president Tayyip Erdogan commented that "there is now a new reality in Syria, politically and diplomatically".
The United Nations' envoy to the country told reporters that the need for an orderly transition in Syria has never been more urgent. Geir Pedersen also called for calm and avoidance of bloodshed.
