Nabil Elaraby, the Arab League’s secretary general, called on Mr Assad to resign immediately, and offered his family safe passage out of Syria.
However, he gave no further details on his proposal at an Arab League foreign ministers’ meeting in Doha, Qatar, on Monday morning.
The League also promised $100 million (£64 million) for Syrian refugees in neighbouring countries.
Tunisian President President Moncef Marzouki also offered Assad asylum in February if it would end the conflict.
Assad has shown little inclination to step down. On Sunday, his forces attacked remaining rebel pockets of resistance in the capital Damascus
While government forces claimed to have quelled attacks in the capital, Damascus, fierce street battles erupted in the Saladin district of Aleppo, a city of three million people that serves as Syria’s main northern commercial hub.
“The sound of bombardment has been non-stop since last night. For the first time we feel Aleppo has turned into a battle zone,” a housewife told the Reuters news agency by phone from Aleppo.
Syrian forces despatched both troops and armoured vehicles into Saladin district, which is understood to have been under rebel control for the past two days.
The fighting in Aleppo was a sign that the country’s rebel factions are determined to continue pressing home their military initiative after Thursday’s bombing in Damascus, which killed four senior figures in President Assad’s security apparatus.
An alliance between the Alawite Assad family and the predominantly Sunni merchant class of Aleppo is understood to have fractured over recent months. Support for the president has also weakened among powerful local tribal chiefs, whose followers often have access to weapons.
The fighting continued early on Friday, with at least three people were killed when Syrian army helicopters fired rockets at the southeastern Damascus neighbourhood of Saida Zeinab, opposition activists said.
“One rocket hit a three story building. I got through to my relatives in Saida Zainab and could hear the sound of the bombardment on the phone,” Reem, one of the activists, told Reuters by telephone from Damascus.
The civil war escalated dramatically in the past week as rebels closed in on the capital Damascus and launched their most serious blow yet on Assad’s inner circle, killing three top aides in an assassination.
Speaking during a visit to Afghanistan, Mr Cameron said: “I have a very clear message for president Assad. It is time for him to go.
“It is time for transition in the regime. If there isn’t transition it’s quite clear there’s going to be civil war.”
Syria’s opposition proclaimed “the beginning of the end” for Bashar al-Assad’s regime on Wednesday night after a bomb attack in the heart of Damascus killed three of the president’s closest lieutenants, including his powerful brother-in-law.
Striking the very core of Mr Assad’s defence apparatus, the bomb exploded inside the headquarters of Syria’s national security council as officials convened a meeting of the ‘crisis cell’ set up to crush the 16-month uprising against the president’s rule.
For weeks, Mr Assad’s power had been visibly crumbling, weakened by a series of high-profile defections. But yesterday’s attack amounted to an evisceration of his inner sanctum, a solitary strike more devastating than any other rebel act during the past 16 months of blood-letting.
“Mortar shelling resumed in the early morning,” said the Local Coordination Committees, a grassroots network of activists.
The LCC added that regime troops and rebel fighters of the Free Syrian Army also clashed in the Damascus district of Kfar Sousa.
The Britain-based Syrian Observatory for Human Rights reported “dawn battles on the road south of Kfar Sousa, between rebel fighters and soldiers who were in a convoy passing through the area.”
“I did not sleep all night,” a resident of nearby Jaramana told AFP. “It was a real war zone.”
Speaking on condition of anonymity, he said he heard “rockets and shooting until the early hours of the morning,” and that he saw four armoured vehicles heading towards the embattled areas.
The British-based Syrian Observatory for Human Rights said that Friday’s shelling in the city of Houla, in central Homs province, had led to what seemed to be one of the bloodiest episodes so far of the 15-month long uprising.
Amateur videos posted on YouTube showed images of children lying dead on a floor. Some of their corpses badly mangled, including at least one child whose head had been partly blown away.
The latest flare-up of violence came as Kofi Annan, the United Nations-Arab League envoy to Syria who brokered a repeatedly violated ceasefire last month, finalised plans to return to Damascus.
The fresh claims of regime atrocities led the opposition Syrian National Council to call on the UN Security Council to take urgent action. At the same time, the Observatory accused the Arab and international communities of being “complicit” in the killing, saying that shelling that had begun on Friday had continued into the night. The Observatory said the international community was standing “silent in the face of the massacres committed by the Syrian regime.”
The explosions shortly before 8am on Thursday morning took place near a busy junction as children were on their way to school and employees on their way to work, the reports said.
Within minutes, gruesome pictures and video were posted online showing body parts in the streets and a mangled wreckage of cars on fire, their drivers still inside. One showed a woman’s burned hand still wearing gold bracelets clutching a steering wheel.
Videos posted online from flashpoints across major cities from Damascus to Idlib showed that the ceasefire dictated by the Annan peace deal had encouraged more people than recently to join protests after Friday prayers.
But the so-called “Day of Revolution for all Syrians” failed in its attempts to pose an overwhelming threat to the Assad regime, as activists hoped.
Regime forces were able to limit the numbers and their effect using force, but without sufficient violence to call the Annan plan into question.
The Local Co-ordination Committees website said there had been 31 incidents of gunfire being turned on protests, and about 11 deaths, fewer than on other Fridays.
Omar al-Khani, from Qaboun in Damascus, one of the places named, said forces had fired on the funeral of Fares al-Habboul, a young man killed on Thursday, also in breach of the ceasefire.
The Foreign Secretary said the “reality” was that some governments – although not Britain’s – would supply weapons to Mr Assad’s enemies if this “best hope” for peace failed.
Saudi Arabia and Qatar, Mr Assad’s leading Arab critics, have publicly favoured arming the opposition. They joined 82 countries and international organisations in Istanbul for the second meeting of the “Friends of Syria” group.
The aim of the gathering, convened by Turkey’s government, was to escalate the pressure on Mr Assad and rally support for a six-point peace plan devised by Kofi Annan, the former United Nations secretary general.
This proposes a ceasefire monitored by UN observers followed by “Syrian-led” negotiations to settle the crisis, with free access for aid agencies and the media. Crucially, Mr Annan’s plan stops short of demanding Mr Assad’s resignation.
Syria’s regime accepted the proposals last Tuesday, but Mr Assad has yet to order a ceasefire or withdraw troops and tanks from urban areas. On the contrary, the bloodshed has continued with dozens more reported to have been killed yesterday.
Artillery and anti-aircraft gun barrages hit the suburbs of Harasta and Irbin, retaken from rebels by Assad’s forces two months ago, and army helicopters were heard flying over the area, on the eastern edge of the capital, the activists said.
Assad’s forces reasserted their control of Damascus suburbs in January after days of tank and artillery shelling that beat back rebels and lessened street protests against the 42-year rule of Assad and his father, the late President Hafez al-Assad.
The suburbs are a linked series of towns inhabited mostly by members of Syria’s Sunni Muslim majority, who have grown increasingly resentful at the domination of the Assads, who belong to the minority Alawite sect, an offshoot of Islam.
The Damascus assault and rebel fighters’ flight on Tuesday from the eastern city of Deir al-Zor mark the latest setbacks for the armed opposition, which also faced accusations of torture and brutality from a leading human rights body.
But as Assad made advances on the ground, he appeared to suffer a setback on the diplomatic front, with key-ally Moscow adopting a new, sharper tone after months of publicly standing by his government.