The army launched its all-out assault last Saturday, subjecting rebel-held areas to continuous bombardment.
"The most heated battles are taking place on the Khaldiyeh and Bab Hud fronts," Homs-based activist Yazan told AFP via the Internet.
"Starting Friday, regime troops started to advance slowly, seizing several buildings on the edges of Khaldiyeh district," the activist said.
Syria's state news agency SANA said the army captured several buildings in Bab Hud in the city centre.
"The situation is very difficult here. If nothing changes, Homs will fall," Yazan said.
The activist also reported "the collapse of several buildings" because of the shelling, adding that an unknown number of civilians were trapped under the rubble.
Warplanes also launched air raids on the city dubbed "the capital of the revolution" by anti-Assad activists, said the Syrian Observatory for Human Rights.
It reported shelling of the already damaged, landmark mosque of Khaled Ibn al-Waleed.
In protest at the assault, representatives of Homs' opposition city council today suspended their membership of the dissident National Coalition in Istanbul, Turkey.
Lebanese Shiite Hezbollah and the authorities in Tehran are staunch backers of Assad's regime.
"They are using weapons of mass destruction against Homs, while the international community does nothing," Shemali told AFP by telephone.
Yesterday Rupert Colville, spokesman for UN High Commissioner for Human Rights Navi Pillay, expressed extreme concern "about the human rights and humanitarian impact" of the regime assault on Homs.
The UN believes some 2,500 to 4,000 people are trapped in the besieged areas, and urges "unrestricted and immediate humanitarian access to all conflict-affected populations in besieged areas".
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