Calculations of the formation of the cathode layer of an self-sustained high-pressure volume discharge with pre-ionization of the gas medium excited by nano-and subnanosecond voltage pulses are carried out. It is shown that at pressures of ~ 1 atm at the final stage of the cathode layer formation, conditions for the generation of escaping electrons arise. We also considered the transition of electrons into runaway mode from the area of electric field amplification in front of the plasma (streamer) channel, which originates from the top of the micro-spike on the cathode. It is shown that at pressures of ~ 10 atm, electrons can transfer into runaway mode immediately after emission from the top of micro-spike in amplified electric field. Thus obtained runaway electrons can create pre-ionization of the gaseous medium and provide the formation of the initial phase of the discharge in volume form in the system without external pre-ionization.