Syria's state news agency claimed "sleeper cells" were publishing clips on social media from public areas of Damascus to suggest they had taken control of them "with the aim of spreading chaos among citizens".
The Syrian government meanwhile denied rumours that Assad had fled the city.
Moreover, the interior minister has said there was a "very strong" military cordon around the capital.
But government forces have notably failed to provide any such defence in the cities, towns and villages that have fallen to rebel factions.
Rebel forces led by the Islamist group Hayat Tahrir al-Sham (HTS) have made a lightning advance over the past fortnight after launching a surprise offensive, capturing the north-western cities of Aleppo and Hama before continuing south and entering parts of the strategically important city of Homs.
Their progress has sparked an uprising by allied rebels in the southern region of Daraa, who have taken control of parts of the region.