Syrian forces have kept up a day-long ground and air attack against rebels in parts of the city of Aleppo.
The BBC's Ian Pannell, who is in Aleppo, has seen fierce battles, with a number of rebel fighters killed.
Rebels with the Free Syrian Army (FSA) say they have repelled an army incursion and destroyed tanks, but there is no independent verification.
Western nations have warned of a potential massacre in Aleppo, Syria's most populous city.
There are also reports of fighting in the western city of Homs, where state media said a number of rebels had been killed, and in the provinces of Hama and Deraa.
Activists said Syrian tanks began moving in on south-western districts of Aleppo city early on Saturday.
The bombardment of rebel-held areas intensified throughout the day, with military aircraft flying over at low altitudes, they said, and violent clashes were reported around the Salah al-Din and Hamdanieh quarters near the centre.
Our correspondent says there has been constant shelling and mortar rounds all day, with helicopter gunships deployed.
The rebels in Aleppo are upbeat but vastly outgunned and outmanned by forces loyal to Syrian President Bashar al-Assad, he adds.
A government official told the AFP news agency: "Rebels are stationed in narrow streets, in which fighting will be difficult."
Both sides are braced for heavy casualties.
The Syrian Observatory for Human Rights said 29 people were killed in Aleppo on Saturday. They were among at least 90 people to die across the country throughout the day.
Syrian state television said that rebels, having failed in Damascus, were now trying to turn Aleppo into a den for their terrorism.
On Friday, the Red Crescent suspended some of its operations in Aleppo because of the heavy fighting, which began more than a week ago.
A steady stream of vehicles has been heading out of the city carrying hundreds of families trying to escape the violence and deteriorating conditions.
Tens of thousands of people have fled Syria over the past few months. Jordan, to the south, is preparing to open its first official camp to house some of the 140,000 Syrians who its say have crossed its border.
Russia, an ally of Syria, warned of a "tragedy" in Aleppo, saying international support for the rebels would lead to "more blood".
Russian Foreign Minister Sergei Lavrov said the Syrian government could not be expected to "stand by" when rebels were occupying Aleppo and other areas and that the West and Syria's neighbours were "essentially encouraging, supporting and directing an armed struggle against the regime".
Later, Mr Lavrov also denied speculation that Moscow would offer asylum to Mr Assad if he chose to leave Syria.
"We have said more than once publicly that we are not even thinking about this," he told reporters.
The fighting comes after two weeks during which rebels made significant gains.
On 18 July, an attack at Syrian security headquarters in Damascus killed four senior officials, including the defence minister and President Assad's brother-in-law.
The Free Syrian Army (FSA) took control of several parts of Damascus before being driven out by a government counter-offensive. The rebels also seized several border crossings with Turkey and Iraq.
Earlier this week, thousands of government forces were moved from the border with Turkey to join fierce fighting in Aleppo, activists said.
On Friday, UN Secretary General Ban Ki-moon urged the Syrian government to halt its offensive and demanded a clear statement that chemical weapons would not be used under any circumstances.
Syria has implicitly acknowledged that it has chemical weapons but says it will not use them against its own people, only against foreign invaders.
The former head of the UN monitoring mission in Syria, Maj Gen Robert Mood, said it was "only a matter of time" until President Assad was ousted.
On Saturday, the Syrian Observatory said that more than 20,000 people, including civilians, rebels, government soldiers and government figures, since the uprising began in March last year.
The UN said in May that at least 10,000 people had been killed.
In June, the Syrian government, which blames the violence on foreign-backed "armed terrorist gangs", reported that 6,947 Syrians had died, including at least 3,211 civilians and 2,566 security forces personnel.