O'Brien also pointed to the some 700,000 people living in other besieged areas across war-ravaged Syria.
"The intensity of attacks on eastern Aleppo neighbourhoods over the past few days has forced thousands of civilians to flee to other parts of the city," UN humanitarian chief Stephen O'Brien said in a statement.
He pointed to reports from the UN's humanitarian partners on the ground indicating that "up to 16,000 people have been displaced, many into uncertain and precarious situations".
"It is likely that thousands more will have no choice but to flee should fighting continue to spread and intensify over the coming days," he warned.
His comments came as the Syrian army advanced deep inside east Aleppo, taking several neighbourhoods from the opposition in an onslaught to recapture the entire city.
Many of those fleeing the intense battles in eastern Aleppo have gone to districts held by the government or Kurdish forces, others heading south into areas still under opposition control.
O'Brien voiced concern over the "deeply alarming and chilling situation unfolding" across all of Aleppo.
The situation in eastern Aleppo is dire, with intensified ground fighting and indiscriminate aerial bombardment reportedly killing and injuring many civilians, he said.
"There are no functioning hospitals left, and official food stocks are practically finished".
At the same time, indiscriminate shelling on government-held western Aleppo has killed and injured civilians and has displaced more than 20,000 people in recent weeks, he said.
O'Brien said the UN and its partners were trying their best to help those displaced by the fighting across the city, and stood ready to bring aid into eastern Aleppo and to carry out medical evacuations if they managed to get access.
"The parties to the conflict in Syria have shown time and again that they are willing to take any action to secure military advantage even if it means killing, maiming or starving civilians into submission in the process," he said.
"In these areas, as in eastern Aleppo, there is no protection and little access to life-saving items," he said, warning that "people in these besieged areas are trapped, terrified and running out of time."
O'Brien urged all parties to the conflict "to restore basic humanity in Syria," insisting that "the people of Syria have suffered far too much and for far too long."