Plumes of smoke were seen rising from the suburbs as the army sought to drive out the Free Syrian Army.
The fighting came as UN and Arab League envoy Kofi Annan prepared to meet Russia’s foreign minister for talks on the Syrian crisis.
Russia has been backing Syrian President Bashar al-Assad’s regime.
Mr Annan is expected to urge Foreign Minister Sergei Lavrov to put pressure on the Syrian authorities to begin a political transition, although BBC Moscow correspondent Steve Rosenberg says there is little sign that the Kremlin is ready to do that.
The International Committee of the Red Cross (ICRC) has said the conflict in Syria is now in effect a civil war.
It means combatants across Syria are now subject to the Geneva Conventions and could be liable for war crimes prosecution in the future.
The ICRC had previously regarded only the areas around Idlib, Homs and Hama as warzones.
The BBC’s Jim Muir says clashes between government forces and Free Syrian Army rebels seem to be creeping ever closer to the heart of Damascus and the centre of the regime’s power.
Tanks and mortars were reportedly used on the southern edge of the city, in areas like Tadhamon and Midan and around nearby Palestinian refugee camps.
Residents were said to be fleeing some areas, while in other parts of the city protesters blocked motorways with burning tyres.
The government has denied that it had used heavy weapons in its attack on the village of Tremseh on Thursday.
UN observers arrived in Tremseh as villagers claimed military forces had carried out indiscriminate attacks
Activists initially described fighting in Tremseh, which is near the city of Hama, as a massacre of dozens of civilians, but later accounts suggested most of the dead were armed rebels.
UN observers at the scene have said Syrian forces used heavy artillery, tanks and helicopters, but Damascus denies those allegations and said just two civilians had been killed.
The accusations, if proved, would mean Damascus had broken an agreement it made with UN and Arab League envoy Kofi Annan.
Further pressure was put on the government of President Bashar al-Assad when the International Committee of the Red Cross, which oversees the Geneva Conventions, said fighting had now spread beyond the three hotspots of Idlib, Homs and Hama.
Spokesman Hicham Hassan said Syria was now regarded as a “non-international armed conflict”, which is the technical term for civil war.
“What matters is that international humanitarian law applies wherever hostilities between government forces and opposition groups are taking place across the country,” he said.
The BBC’s Imogen Foulkes in Geneva says the statement is significant because it is the Red Cross’ job to monitor the conduct of the fighting, and to tell warring parties what their obligations are.
Under the Geneva Conventions, indiscriminate attacks on civilians, attacks on medical personnel or the destruction of basic services like water or electricity are forbidden and can be prosecuted as war crimes.
From now on, all those fighting in Syria are officially subject to the laws of war, and could end up at a war crimes tribunal if they disobey them.
The ICRC’s announcement echoes both the UN’s head of peacekeeping Herve Ladsous and President Assad, who has said the country is at war.
Some 16,000 people are thought to have been killed since the uprising against Mr Assad’s regime began in March 2011.
UN diplomats are attempting to agree a way forward for the organisation’s monitoring mission in the country.
The mission’s mandate runs out on Friday, and Western nations are trying to get Russia and China to agree to a strengthened resolution authorising sanctions.
Source: BBC News