Federal law-enforcement authorities in New York are investigating whether Uber Technologies Inc. used software to interfere illegally with a competitor, according to people familiar with the investigation, adding to legal pressures facing the ride-hailing company and its new chief executive.

The investigation, led by the Federal Bureau of Investigation’s New York office and the U.S. attorney’s office in Manhattan, is focused on a defunct Uber program, known internally as “Hell,” that could track drivers working for rival service...