Nanosecond pulse discharges can provide high reduced electric field for exciting high-energy electrons, and the ultrafast rising time of the applied pulse can effectively suppress the generation of spark streamer and produce homogeneous discharges preionized by runaway electrons in atmospheric-pressure air. In this paper, the electrostatic field in a tube-plate electrodes gap is calculated using a calculation software. Furthermore, a simple physical model of nanosecond pulse discharges is established to investigate the behavior of the runaway electrons during the nanosecond pulse discharges with a rise time of 1.6 ns and a full-width at half-maximum of 3–5 ns in air. The physical model is coded by a numerical software, and then the runaway electrons and electron avalanche are investigated under different conditions. The simulated results show that the applied voltage, voltage polarity, and gas pressure can significantly affect the formation of the avalanche and the behavior of the runaway electrons. The inception time of runaway breakdown decreases when the applied voltage increases. In addition, the threshold voltage of runaway breakdown has a minimum value (10 kPa) with the variation of gas pressure.