But a Britain-based human rights watchdog reported that the assault had been repulsed and that clashes were continuing on the outskirts of the neighbourhood.
"The area is under control," the security source in Damascus said, requesting anonymity.
"The army has started combing the area building by building and house by house. Now the troops are searching every basement and tunnel for arms and terrorists.
"There remain only few pockets" of resistance, the source added.
The Syrian Observatory for Human Rights reported, however, that rebel forces were "preventing an attempt to storm" the neighbourhood.
The watchdog said gunfire could be heard in several districts of the city.
Homs-based activist Hadi Abdullah reported clashes and heavy shelling of the neighbourhood but insisted that regime forces had not entered Baba Amr.
"Regime forces did not enter Baba Amr until this moment. They are surrounding the district, while clashes are concentrated in the neighbourhoods of Inshaat and Malaab," he told AFP.
Abdullah said that activists were "evacuating families because shelling has been targeting places that were considered safe in the past."
On Wednesday, elite troops, backed by tanks, had massed around Baba Amr and activists expressed fears of an imminent attack on the neighbourhood which has been a stronghold of rebels of the Free Syrian Army.