KABUL—The United Nations warned of a humanitarian crisis in the northern Afghan city of Kunduz, where it said fighting between the Taliban and U.S.-backed Afghan forces had forced some 10,000 residents to flee their homes.

“Many families were unable to bring their possessions with them and are in a precarious position. We have had reports that some families have been forced to sleep out in the open and many have few food supplies,” Dominic Parker, head of the U.N. Office for the Coordination of Humanitarian Affairs, said on...