Many of them are sheltering in bunkers beneath the city's Azot chemical plant.
The last bridge leading out of the city was destroyed in fighting earlier this week - effectively trapping its 12,000 remaining residents inside.
For weeks capturing Severodonetsk has been a top military goal for Russia, which now controls most of the city.
"The lack of water and sanitation is a big worry. It's a huge concern for us because people cannot survive for long without water," spokesperson for the UN's Humanitarian Affairs office Saviano Abreu told the BBC.
