Syrian government forces have launched a ground assault in the city of Aleppo, and are bombarding rebel-held areas with heavy artillery, activists and witnesses have said.
Rebel forces say they have beaten off the first wave of attacks that began on Saturday morning, but that the battle continues.
At least 186 people were killed in violence in Aleppo on Saturday.
Civilians crowded into basements seeking refuge from the bombing, with the SOHR's Rami Abdel Rahman describing the clashes as the uprising's "fiercest".
"There are thousands of people in the streets fleeing the bombardment. They're being terrorised by helicopter gunships flying at low altitude," said an activist, adding many had taken refuge in public parks.
Battered and burnt-out tanks littered one of the main roads into the commercial capital, according to activists, who said the move marked the beginning of the expected government assault on the city.
Syrian army troops were focussing on the rebel-held neighbourhoods of Saleheddine and Sakhour, the SOHR said.
Colonel Abdel Jabbar al-Oqaidi, a member of the Free Syrian Army, said that the rebels had managed to repulse the government's attacks on Saturday, redirecting them towards the neighbourhood of Hamdaniyeh.
He said his forces had destroyed eight armoured vehicles, and that 100 government tanks were massed on the outskirts of Aleppo.
Abdel Rahman said the fact the soldiers had been stopped in Salaheddine "does not necessarily mean a withdrawal as their strategy is to bombard ... to cause an exodus then launch an assault even more fierce."
Saleheddine has been a stronghold of Free Syrian Army since they seized large parts of Syria's second city on July 20.
Tanks, helicopter gunships and warplanes have poured into Aleppo over the last two days with troops firing on a string of Free Syrian Army neighbourhoods in the battle to control the commercial hub of 2.5 million people.
In the surrounding countryside, however, the balance of power is changing fast. In an exclusive report from Al-Bab in Aleppo province, Al Jazeera's Anita McNaught said Free Syrian Army fighters have taken control while the Syrian army has largely disappeared.
"What we have been hearing from activists is that some neighbourhoods in Aleppo have become battlefields, especially the neighbourhood of Salaheddine in the southwest," Al Jazeera's Zeina Khodr reported from Beirut, in neighbouring Lebanon, on Saturday.
"The battle for Aleppo really has begun. We have seen clashes over the past two days, and we have seen both sides send in reinforcements. It is really important for both sides to win, to take control of Aleppo, it's such a strategic territory."
Al Jazeera's Stephanie Dekker, reporting from Amman, in neighbouring Jordan, said that activists in Aleppo were reporting that the Free Syrian Army were holding strong in neighbourhoods that they held, but that residents there were "terrified" of being killed if the army were to move in.
A medic in Aleppo, speaking on condition of anonymity, told Al Jazeera that parts of the city remained relatively calm, and that violence was only focused on Saleheddine and other areas held by the Free Syrian Army opposition.
Russia warned on Saturday that a "tragedy" was looming in Aleppo but also said it was unrealistic to expect the government would stand by when armed rebels were occupying major cities.
In separate remarks, Lavrov said that Russia was "not even thinking about" offering Assad asylum.
Meanwhile, the International Committee of the Red Cross announced that it was temporarily relocating some of its foreign staff from Damascus to Lebanon, and was partially suspending operations in Aleppo, due to security concerns.
On Friday, activists said helicopter gunships opened fire on civilians in Aleppo province and that the army had massed troops on the city's borders.
In addition to Aleppo, rebels continued to battle with pro-government forces in the capital Damascus on Saturday. Fresh fighting was also reported on Friday in Idlib and Deraa, activists said.
In-depth coverage of escalating violence across Syria
Meanwhile in neighbouring Lebanon on Saturday, gun battles between Sunni Muslims and their Alawite rivals left at least eight people injured in a Sunni district in Tripoli. The city has seen sporadic fighting throughout the Syrian uprising.
The heavy fighting around Aleppo and Damascus marked a new phase of the violence that has gripped Syria since protests broke out about 17 months ago.
According to figures released by the SOHR on Saturday, more than 20,000 people have been killed since the uprising began in March 2011. That number includes 13,978 civilians and rebels, 968 army defectors and 5,082 government soldiers, SOHR Chief Abdel Rahman said.
Source: Al Jazeera And Agencies