The BBC's Jim Muir says fighting has intensified in many parts of Syria
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The Syrian capital Damascus has seen some of the heaviest fighting of the conflict so far, according to reports from activists and residents.
Mortar and small-arms fire was reported in several areas as government forces clashed with the Free Syrian Army.
The fighting came as UN and Arab League envoy Kofi Annan was to urge Russian leaders to put pressure on Damascus.
But Russia said it was being "blackmailed" by the West to get it to back a draft resolution on sanctions.
Speaking ahead of Mr Annan's visit, Foreign Minister Sergey Lavrov said the West had told Russia to back a draft resolution on sanctions or it would "refuse to extend the mandate of the observer mission".
It was "not right" to say that pressure should only be brought on the government of President Bashar al-Assad and not the opposition to end the conflict, he added.
"We do not support Assad," he said. "We support what has been agreed by all sides."
Mr Annan is expected to urge Mr Lavrov to persuade 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.
Clashes in Tadhamon were said to be continuing on Monday morning
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.'Residents fleeing'
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.
Mortars were reportedly used on the southern edge of the city, in areas like Tadhamon and Midan and around nearby Palestinian refugee camps.
Activists said clashes continued into the early hours of the morning.
A convoy of army reinforcements was reported to have been attacked by rebels in Kfar Sousa to the west, leading to further clashes there.
Residents were said to be fleeing some areas, while in other parts of the city protesters blocked motorways with burning tyres.
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Annan's six-point plan
1. Syrian-led political process to address the aspirations and concerns of the Syrian people
2. End to violence by all sides; army troops to stop using heavy weapons and withdraw to barracks
3. Parties to allow humanitarian aid
4. Authorities to free political detainees
5. Authorities to ensure freedom of movement for journalists
6. Authorities to allow peaceful demonstrations
There has been frequent trouble in these areas - barely three miles (4-5km) from the centre - for months.
But as with many of the suburbs ringing the city slightly further out, all the government's repeated efforts to stifle defiance have failed, our correspondent says.
The government has denied that it had used heavy weapons in its attack on the village of Tremseh on Thursday.
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.Geneva conventions
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.
Are you in Syria or do you have relatives there? What is your experience of the unrest in the country?