Shooting was reported in one of the main central streets and a square housing the Central Bank.
There were also reports of tanks in the south-western area of Midan, and clashes involving helicopters in the north and north-east of the city.
The fighting comes as UN-Arab League envoy Kofi Annan prepares to hold talks with Russian President Vladimir Putin about Syria.
Russia is a key ally of Syria and the meeting comes amid mounting pressure for increased international action against the country.
US Secretary of State Hillary Clinton said she hoped Mr Annan could persuade Russia to accept tougher measures.
Meanwhile a key Syrian defector, Nawaf Fares, the ex-ambassador to Iraq, has told the BBC the regime will not hesitate to use chemical weapons if it is cornered.
Mr Fares, the most senior Syrian politician to have defected to the opposition, also said that major bombings across Syria had been orchestrated by the government in collaboration with al-Qaeda.
‘Matter of time’
Violence is continuing to spread across Syria and in the capital Damascus as rebels – now better-equipped and more organised – confront the army and government-backed militia.
The rebel Free Syrian Army has said it has launched “Operation Damascus Volcano”, and has called for an escalation of attacks on regime targets and the blocking of main highways all around the country.
One of the biggest and most organised opposition groups, the Muslim Brotherhood, has called on all Syrians to join what it called a decisive battle.
Witnesses say the government’s military deployment in Damascus is the biggest since the 16-month uprising against the government of President Bashar al-Assad began.
Clashes were reported in a major thoroughfare, Baghdad Street, the first time fighting has reached central Damascus since the conflict began.
Also, machine-gun fire was reported in nearby Sabaa Bahrat square, site of the Central Bank and scene of several major pro-government demonstrations.
Activists reported continued clashes on the south-western side of the city, including in Midan.
“The army is shelling al-Midan hysterically; the collapsing regime has gone mad,” one activist told AFP.
Fighting is also said to have broken out on the other side, at Barzeh and Qaboun. Attack helicopters were seen there firing rockets for the first time since the uprising began.
State media gave little prominence to the fighting, but said security forces were pursuing remnants of a “terrorist gang” in Nahr Eisha.
Mrs Clinton told the BBC’s Kim Ghattas that she hoped Russia could be convinced of the need for a UN resolution mandating tough sanctions against Damascus if it failed to comply with Mr Annan’s peace plan.
Mrs Clinton said the US “was working very hard to try to have a managed transition” but that the clashes in and around Damascus showed that the end of the Assad government was “a matter of time”.
Russian Foreign Minister Sergei Lavrov said on Monday that the West had threatened to end the current UN observer mission in Syria if Moscow opposed its draft resolution, which threatens more sanctions. He said the attempts contained “elements of blackmail”.
Moscow’s UN ambassador Vitaly Churkin said he had made it very clear he would vote against the resolution.
Moscow has circulated its own draft resolution calling for the extension of the mission’s mandate – which ends on Friday – but without a threat of sanctions.
Russia and China have blocked two previous UN resolutions that condemned President Assad’s government for the continuing violence.
Syria is known to have a significant stockpile of chemical weapons. There have been growing concerns in neighbouring countries and among Western governments about the security of such weapons should the government fall.
Asked if he thought President Assad might use chemical weapons against the opposition, Mr Fares told BBC security correspondent Frank Gardner that he would not rule it out, describing Mr Assad as “a wounded wolf and cornered”.
Mr Fares’s claim that Sunni Muslim militants in al-Qaeda are collaborating with a government dominated by those from the minority Allawite sect will surprise many.
Challenged on his view that al-Qaeda was collaborating with the government despite this, Mr Fares said: “There is enough evidence in history that lots of enemies meet when their interests meet.”
He added: “Al-Qaeda is searching for space to move and means of support, the regime is looking for ways to terrorise the Syrian people.”
Mr Fares held senior positions in the ruling Baath Party and powerful security services, and served as governor in several provinces.
“It doesn’t occur to any Syrian, not only me, that Bashar al-Assad will let go of power through political interventions… He will be ousted only by force,” Mr Fares said.
Mr Fares was dismissive of Mr Annan’s visit to Russia, saying success there was “impossible”.
“Several months have passed and… the regime hasn’t implemented a single article of [Kofi Annan's] plan.”