The Free Syrian Army also claimed it had shot down a helicopter in the capital
Rebels declared the battle to "liberate" Damascus has begun as heavy fighting raged across the city and Russia said an agreement is possible for a UN resolution on the Syria crisis.
The proclamation by the Free Syrian Army, which also claimed it had shot down a helicopter in the capital, came as peace envoy Kofi Annan said the 16-month crisis increasingly described as a civil war was at a "critical time."
The opposition Syrian National Council meanwhile said Syrians will seek new ways to confront President Bashar al-Assad if the UN Security Council fails to threaten sanctions, ahead of a vote showdown Wednesday between the major powers.
Heavy machinegun fire was reported in Damascus's Sabaa Bahrat Square, where Assad's regime has staged rallies to counter anti-regime protests that erupted in March 2011.
At least 19 people were killed as tanks and helicopter gunships were deployed in Qaboon district and battles were fought in Al-Midan and Al-Hajar Al-Aswad, said the Syrian Observatory for Human Rights.
As the fighting inched closer to the regime's nerve centre, FSA spokesman Colonel Kassem Saadeddine said "victory is nigh" and the struggle would go on until the city was conquered.
"We have transferred the battle from Damascus province to the capital. We have a clear plan to control the whole of Damascus. We only have light weapons, but it's enough."
"Expect surprises," Saadeddine added, before claiming later that rebels had downed a helicopter over Qaboon, although an activist in the district said there was "no foundation" to the report.
Fighting in the city has raged since Sunday, with the rebels announcing a full-scale offensive dubbed "the Damascus volcano and earthquakes of Syria."
An activist who said he was in Al-Midan neighbourhood said the army was shelling the neighbourhood "hysterically."
"The collapsing regime has gone mad," the man calling himself Abu Musab said via Skype.
"The army has tried to storm the district, but the Free Syrian Army has stopped them. So they have intensified their shelling. They are shelling everything," he said.
AFP could not independently verify the account.
Witnesses also reported heavy machinegun fire in Sabaa Bahrat Square in central Damascus and in nearby Baghdad Street.
But an army officer in Damascus told AFP that troops have "the situation under control" and were "chasing the terrorists seeking refuge in apartments and mosques."
The source said "battles raged" in Qaboon, "where the majority of rebels were," adding that "33 terrorists were killed, 15 were wounded and 145 were arrested," referring to rebels.
In that context, the Israeli army's intelligence chief said Syrian troops had been moved from the Golan Heights towards conflict zones including Damascus.
"Assad has removed many of his forces that were in the Golan Heights to the areas of conflict," Major General Aviv Kochavi told MPs.
In Moscow, Russian President Vladimir Putin told Annan that he would "do everything" to support the UN-Arab League peace envoy's plan to end the conflict.
Annan told Putin "the Syrian crisis is at a critical time."
Later, Russian Foreign Minister Sergei Lavrov said he saw "no reason why we cannot also agree at the UN Security Council. We are ready for this."
Annan's Moscow meetings came one day before Western powers plan to hold a vote on a UN resolution that threatens sanctions against Damascus.
The council must also vote to decide on renewing the 300-strong UN Supervision Mission in Syria, deployed to monitor an April 12 ceasefire Assad agreed with Annan.
UN leader Ban Ki-moon "called on Russia to use its influence to ensure the full and immediate implementation" of Annan's plan in a telephone call with Lavrov, a spokesman said.
Ban was due in Beijing on Wednesday, also on a mission to get support for tougher action on Syria.
Russia and China have twice blocked resolutions against Syria at the Security Council, which remains divided over Western calls to impose new sanctions.
Russia, which has threatened to veto a proposed western resolution calling for sanctions, put forward a new draft on Tuesday which was rejected by Britain, France, United States, Germany and Portugal, diplomats said.
"Barring a last minute surprise, we should now go for a vote on Wednesday and we expect a veto by Russia and China," said the UN envoy of a western nation.
Representatives of the Syrian National Council met ambassadors from the Security Council, including Russian envoy Vitaly Churkin, in a bid to persuade them to back sanctions.
The group warned that it will seek alternative ways to defend civilians if the Security Council does not threaten sanctions over the 16-month-old conflict.
On a visit to Syria's neighbour Jordan, British Foreign Secretary William Hague said the crisis is too unpredictable to rule out "any option."
Stepping up the pressure, French President Francois Hollande said "the Russians must understand that they cannot be seen as the only ones or almost the only ones hindering the search for a solution."
The Observatory said at least 78 people were killed across Syria on Tuesday -- 43 civilians, 23 members of the security forces and 12 rebels -- adding to its toll of more than 17,000 people dead since the uprising began.
Meanwhile a key defector, General Manaf Tlass, a childhood friend of Assad, said in a statement sent to AFP that he was in Paris and called for a "constructive transition" in the country. President Hollande confirmed that Tlass was in the French capital.