Russia's military has asked President Vladimir Putin for permission to resume airstrikes on the eastern part of the city after a lull in fighting last week, Russia's state news agency Tass reported Friday.
Kremlin spokesman Dmitry Peskov said Mr Putin did not consider it appropriate to restart the strikes, and said the humanitarian pause should continue.
Prior to the war, Aleppo was Syria's largest city and commercial capital.
At least 15 civilians have been killed in heavy shelling and suicide vehicle bomb attacks - among them a number of children, according to the UK-based Syrian Observatory for Human Rights.
In parallel, the regime has fired artillery targeting the neighbourhoods of Tariq al-Bab, Al-Sokhour and Masaken Hanano, east of Aleppo, besieged by the army and held by opposition forces.
The military media also refuted the rumors which reported that the terrorist groups captured a number of Hezbollah fighters, stressing that the reports are baseless and aim at boosting the morale of the takfiris.
He then stated that the government and Russian Federation were willing to reinstate a cessation of hostilities if rebel forces guaranteed the safety of civilians and those wishing to leave Aleppo through so-called exit corridors.
Rebel forces issued a call-to-arms spanning the entire Syrian opposition to the Assad regime in hopes of breaking a months-long siege that the Syrian army has imposed on the city in an effort to force rebels to surrender. The Syrian army claimed it repelled all rebel attacks without losing any territory.
Syrian Foreign Minister Walid al-Muallem vowed on Friday that the state military would take full control of the war-torn city of Aleppo.
IS trickling out of Mosul as losses mount: US general
There are between 3,500 and 5,000 ISIS militants in Mosul and up to another 2,000 in the broader area, according to US estimates. Moreover, the US-led jets are also taking part in the operation.
Aircraft operated by Russian Federation, a vital ally of Assad, have heavily bombarded rebels in the city, which has been divided between rebel areas and government-held neighborhoods in the west since 2012.
Jabhat Fatah al-Sham, an anti-Assad force linked to al-Qaeda, said it carried out at least one suicide bombing. Syria and Russian Federation claim they are seeking to fight "terrorists", and the West and its partners have been unable to draw clear distinctions between the militias they back and the Islamist forces.
The new offensive was a strong sign that rebel groups vetted by the United States were continuing their tactical alliances with groups linked to Al Qaeda, rather than distancing themselves as Russian Federation has demanded and the Americans have urged.
Rebels, including some armed units formerly loyal to Al Qaeda, broke the siege on Aleppo late in the summer only to watch the regime, with the help of hundreds of elite fighters from Lebanon's Hezbollah, reimpose it shortly afterward.
The area around the Hmeimim air base near the coastal city of Latakia - the main airport used by the Russian air force - was also reportedly targeted, the SOHR said.
The government has maintained a siege on the rebel-held eastern quarters of Aleppo since September, and the United Nations estimates 275,000 people are trapped inside with dwindling supplies of food and medicine.
"All the revolutionary factions, without exception, are participating in the battle", the military spokesman for the Fastaqim faction inside the city said, AP news agency reports.
Aleppo has emerged as a key focal point and symbol of the Syrian war.
A quarter of a million people are living under siege in eastern Aleppo.