Activists said 19 civilians and rebel fighters were killed on Sunday as the soldiers in buses and armoured personnel carriers moved in at dawn, along with at least 50 tanks and armoured vehicles.
The forces of President Bashar al-Assad pushed into the Ghouta area on the eastern edge of Damascus to take part in an offensive in the suburbs of Saqba, Hammouriya and Kfar Batna.
Tanks advanced into the centre of Saqba and Kfar Batna, the activists said, in a move to flush out fighters who had taken over districts just a few kilometres from Assad's centre of power.
"It's urban war. There are bodies in the street," said one activist, speaking from Kfar Batna. Activists said 14 civilians and five insurgents from the rebel Free Syrian Army were killed there and in other suburbs.
The escalating bloodshed prompted the Arab League to suspend the work of its monitors on Saturday. Arab foreign ministers, who have urged Assad to step down and make way for a government of national unity, will discuss the crisis on February 5.
Arab League chief Nabil Elaraby left for New York where he will brief representatives of the U.N. Security Council on Tuesday to seek support for an Arab peace plan that calls on Assad to step aside after 10 months of protests.
He will be joined by Qatari Prime Minister Sheikh Hamad bin Jassim al-Thani, whose country heads the League's committee charged with overseeing Syria.
Speaking shortly before he left Cairo on Sunday, Elaraby said he hoped to overcome resistance from Beijing and Moscow over endorsing the Arab proposals. "There are contacts with China and Russia on this issue," he said.
A Syrian government official was quoted by state media as saying Damascus was surprised by the Arab League decision to suspend monitoring, which would "put pressure on (Security Council) deliberations with the aim of calling for foreign intervention and encouraging armed groups to increase violence."
Assad blames the violence on foreign-backed militants.
State news agency SANA reported funerals on Saturday for 28 soldiers and security force members killed by "armed terrorist groups" in Homs, Hama, Deraa, Deir al-Zor and Damascus province.
Another 24 soldiers were reported killed on Sunday. SANA said six soldiers were killed in a bombing southwest of Damascus, while the opposition Syrian Observatory for Human Rights said 18 soldiers were killed in two separate attacks by army deserters in the northern province of Idlib.
Faced with mass demonstrations against his rule, Assad launched a military crackdown to subdue the protests. Growing numbers of army deserters and gunmen have joined the demonstrators, increasing instability in the country of 23 million people at the heart of the Middle East.
The insurgency has been gradually approaching the capital, whose suburbs, a series of mainly conservative Sunni Muslim towns bordering old gardens and farmland, known as the al-Ghouta, are home to the bulk of Damascus's population.
One activist reported heavy shelling in the suburb of Saqba, and said the army was facing stiff opposition from rebels.
Another, who identified himself as Raid, said mosques had been turned into field hospitals and were appealing for blood supplies. "They cut off the electricity. Petrol stations are empty and the army is preventing people from leaving to get fuel for generators or heating," he said.
The Damascus suburbs have seen large demonstrations demanding the removal of Assad, a member of the minority Alawite sect, an offshoot of Shi'ite Islam that has dominated the mostly Sunni Muslim country for the last five decades.
In Rankous, 30 km (20 miles) north of Damascus by the Lebanese border, Assad's forces have killed at least 33 people in recent days in an attack to dislodge army defectors and insurgents, activists and residents said on Sunday.
Rankous, a mountain town of 25,000 people, has been under tank fire since Wednesday, when several thousand troops laid siege to it, they said.
France, which has been leading calls for stronger international action on Syria, said the Arab League decision highlighted the need to act.
"France vigorously condemns the dramatic escalation of violence in Syria, which has led the Arab League to suspend its observers' mission in Syria," the Foreign Ministry said.
"Dozens of Syrian civilians have been killed in the past days by the savage repression taken by the Syrian regime ... Those responsible for these barbarous acts must answer to their crimes," it said.
The Arab League mission was sent in at the end of last year to observe Syria's implementation of the peace plan, which failed to end the fighting. Gulf states withdrew monitors last week, saying the team could not stop the violence.
The United Nations said in December more than 5,000 people had been killed in the protests and crackdown. Syria says more than 2,000 security force members have been killed by militants.
On Friday, the U.N. Security Council discussed a European-Arab draft resolution aimed at halting the bloodshed. Britain and France said they hoped to put it to a vote next week.
Russia joined China in vetoing a previous Western draft resolution in October, and has said it wants a Syrian-led political process, not "an Arab League-imposed outcome" or Libyan-style "regime change.