Many civilians fear fleeing because of ISIS snipers and landmines, but 157,000 have reached a reception and transit centre since the Iraqi government offensive on west Mosul began a month ago, it said.
"The worst is yet to come, if I can put it this way. Because 400,000 people trapped in the Old City in that situation of panic and penury may inevitably lead to the cork popping somewhere, sometime, presenting us with a fresh outflow of large-scale proportions," said Bruno Geddo, the U.N. High Commissioner for Refugees (UNHCR) representative in Iraq.
Fighting in the past week has focused on the Old City, with government forces reaching as close as within 500 meters of al Nuri mosque, from where Islamic State leader Abu Bakr al-Baghdadi declared a caliphate spanning parts of Iraq and Syria in July 2014.
Islamic State militants are now on the back foot, with their stronghold in Syria also under attack. But they still hold an estimated 40 percent of western Mosul and the battle to recapture it could take weeks.
Civilians are streaming out at an increasing rate, now averaging 8,000-12,000 per day who reach a reception and transit centre at Hammam al-Alil, Geddo said, speaking from that site 20 km (15 miles) south of Mosul where they undergo security screening.
"We also heard stories of people running away under the cover of early morning fog, running away at night, of trying to run away at prayer time when the vigilance at ISIS (Islamic State) checkpoints is lower," he said.
The displaced told stories of a "dramatic" situation in the Old City, amid shortages of food, fuel and electricity, he said.
"People have started to burn furniture, old clothes, plastic, anything they can burn to keep warm at night, because it is still raining heavily and the temperatures at night in particular drop significantly."
The government halted offensive operations on Thursday due to cloudy weather, which makes it difficult to bring in air support.
In Mosul, Lieutenant Colonel Abdul Ameer said the government forces were ready to start a big assault but were waiting for the weather to improve. He also said the militants' ability to send out car bombs had significantly diminished after the security forces sealed off most roads inside Old City.
"The more you go without food, the more you become panicked and the more you want to run away. At the same time it (the outflow) is increasing because the security forces are advancing and therefore more people are in a position to run away where the risk is likely more mitigated," Geddo said.