There have been fresh reports of clashes and attacks by helicopter gunships in Aleppo as the Syrian army offensive enters its fourth day.
State TV said Syrian forces were inflicting heavy losses on "terrorist groups" in Aleppo and also claimed government successes in Homs.
Thousands of refugees are continuing to flee Aleppo - Syria's commercial hub.
Meanwhile the US and Turkey have agreed to step up efforts to achieve "political transition" in Syria.
This would include the departure of Syrian President Bashar al-Assad, the White House said.
According to Syrian TV, government troops have been gaining ground in several Aleppo neighbourhoods.
On Monday, it reported that the army had won complete control over Salah al-Din, one of the areas of Aleppo where rebel fighters from the Free Syrian Army had become entrenched.
Analysts say Salah al-Din provides an important route for Syrian troop reinforcements coming from the south.
But activists and rebel commanders denied that government forces had gained ground there.
They report renewed heavy shelling and helicopter gunship attacks there and in other rebel-held areas on Tuesday, especially around the Sakhour quarter on the north-east side of Aleppo.
Fierce fighting also taken place near the headquarters of the feared Air Force Intelligence agency to the west of the city, they say.
State television completely ignored events in Aleppo in its early morning news programmes on Tuesday, the BBC's Jim Muir in Beirut says.
Instead, it broadcast a detailed report from Homs, where it said the quarter of Qarabis, which has long been held by rebel fighters, had finally fallen to government forces.
But later bulletins carried reports of Syrian forces inflicting heavy losses on ''terrorist groups'' in Aleppo districts.
Correspondents say neither side can afford to lose Aleppo, Syria's largest city and is the country's industrial and financial centre.
Until this month it had been spared the daily bloodshed seen in other cities since the uprising began in March 2011.
Supplies are now reported to be running out in certain areas and those citizens remaining in the city face long queues for bread.
Melissa Fleming, a spokeswoman for the UN refugee agency, the UNHCR, said thousands of frightened residents had sought shelter from the fighting in schools, mosques and public buildings.
"These are the people who haven't fled the city, haven't had the means or feel that it's too dangerous to make that journey," she said.
"There are 32 schools in Aleppo that we've identified and in each of those schools, we have heard, or we have witnessed, that 250-300 people were packed inside, many of these families, kids."
In a phone call on Monday, US President Barack Obama and Turkish Prime Minister Recep Tayyip Erdogan agreed to help the growing numbers of refugees - both inside Syria and in neighbouring countries.
Turkey has been a staunch critic of President Assad and has given refuge to large numbers of army defectors.
Thousands of Syrian refugees are also living in refugee camps on the Turkish side of the countries' long border.
Those arriving from Aleppo in recent days have spoken of incessant shelling.
As the crisis deepened, Iran - a close ally of Damascus - warned Turkey not to intervene militarily, the Syrian state-controlled al-Watan newspaper said.
"Any attack on Syrian territory will meet with a harsh response, and the Iranian-Syrian mutual defence agreement will be activated," it reported on Monday.
The newspaper said Turkey and the US were planning to create "a safe haven guarded by the armed gangs" in the north of Syria.
"Turkey has received very strong warnings in the past few hours and the following message - beware changing the rules of the game," al-Watan said.
Turkey has said it will use troops if necessary to prevent another Halabja - a reference to a notorious massacre in Saddam Hussein's Iraq.
Greek officials were due to reveal further details of new border security plans on Tuesday, aimed at stopping Syrian refugees crossing from Turkey.
On Monday, Greece - one of the busiest entry points for illegal immigrants entering the EU - said it was quadrupling the number of guards along its border with Turkey.
In New York, UN Secretary General Ban Ki-moon repeated that he was particularly concerned by the continued use of heavy weapons by Syrian government forces.
He also called on the Syrian government to renounce the possibility of using chemical weapons under any circumstances, and to secure its stockpiles.
Damascus has implicitly acknowledged its possession of chemical weapons but said it would not use them against its own people, only against foreign invaders.