Syria's opposition warned Assad that he would be toppled unless he ushered in democratic reforms, although the UN Security Council failed to agree on a condemnation of the violence.
And in a fresh blow to the regime, 233 members of Syria's ruling Baath party announced their resignation in protest at the deadly crackdown on protesters, according to lists seen by AFP.
"The security services have demolished the values with which grew up. We denounce and condemn everything that has taken place and announce with regret our resignation from the party," they said in a signed statement.
On the international scene, influential US Senator John McCain said Assad has "lost his legitimacy" and called for UN sanctions to force him to halt attacks on his people.
"I obviously think he has lost his legitimacy. He has ordered his army to fire on his own people, and yes I think he should leave," the senator told AFP in Paris.
The Security Council, however, failed to agree on a statement condemning the killing of Syrian protesters, diplomats in New York said. After talks ended in deadlock, Western nations called for an immediate open meeting.
Russia and China blocked the statement proposed by Britain, France, Germany and Portugal that would have condemned the violence and backed calls for an independent investigation.
Germany's defence minister earlier demanded that the Council send a "strong signal" to show international alarm over the unrest in Syria.
The European Union, meanwhile, is mulling sanctions and the UN human rights body has called for a special session in the wake of the Syrian regime's bloody crackdown on pro-democracy protesters.
Five EU countries are also summoning Syria's ambassadors over the violent crushing of dissent, France said, adding it was joined by Britain, Germany, Italy and Spain.
According to human rights activists, the military assault on Daraa, 100 kilometres (62 miles) south of Damascus, has left more than 30 people dead since Monday, with at least 453 civilians killed across Syria since protests first erupted in mid-March.
A military source, meanwhile, said soldiers on Wednesday confronted "terrorist armed groups" who had cut off roads and opened fire on passers-by in a Daraa drive-by shooting.
"One member of the armed forces was martyred and five others were wounded," said the source, quoted by the official media, adding that several of the gunmen were also killed.
He denied satellite television reports of a rift in army ranks.
As the assault on Daraa, an agricultural town near the Jordanian border, entered its third day, the newly formed National Initiative for Change (NIC) warned Assad to institute real democratic reforms or risk "violence, chaos and civil war."
"Either the ruling regime leads itself in a peaceful transition towards democracy ... or it will go through a process of popular protests that will evolve into a massive and grassroots revolution," an NIC statement said.
"If the Syrian president does not wish to be recorded in history as a leader of this transition period, there is no alternative left for Syrians except to move forward along the same path as did the Tunisians, Egyptians and Libyans before them," added the NIC, an umbrella group of more than 150 opposition activists in Syria and abroad.
Syrian protesters took to the streets in even greater numbers after Assad scrapped nearly five decades of draconian emergency rule and abolished the repressive state security court a week ago.
Testing his promised reforms, they staged protests across Syria on Friday, demanding the end to the Baath's grip on political power, the release of political prisoners and the right to protest freely.
However, the security forces cleared demonstrations with tear gas and live rounds, with scores reported killed and hundreds arrested.
On Monday, between 3,000 and 5,000 troops backed by tanks and snipers swept into Daraa, the epicentre of the protests killing at least 25 people, according to rights activists. At least another six people died on Tuesday.
The army said troops entered Daraa "in response to calls for help" from citizens to rid them of "extremist terrorist groups" behind a spate of killings and sabotage.
Security forces also deployed in the northern Damascus suburb of Douma on Monday.
Austria said steps were being taken to evacuate its nationals from Syria, while Syrian ally Turkey said it is sending envoys Thursday to Syria to press for reform.