Mayor Arturo Garino said he was told the following information about the incident.
A southbound vehicle at the port was asked by a U.S. Customs and Border Protection officer to stop, but the driver refused and attempted to run over the officer.
The driver was shot by a CBP officer or officers.
The vehicle's momentum carried it over the border onto the Mexican side of the buffer zone between the two countries.
Officers of Mexico's border agency reportedly surrounded the vehicle and dragged a man out who is dead.
The Nogales International reported that Mexican authorities said two men were in the vehicle, a pickup truck. The driver was reportedly shot in the head and the unharmed passenger was arrested, the newspaper reported.