In this paper, we present the results of numerical investigations into the influence of preionization on the properties of extreme ultraviolet (EUV) lasers based on nanosecond capillary discharges. Prior to application of the main current pulse, gas inside the capillary is usually preionized by a separate current pulse with longer duration and lower amplitude, which creates plasma with minimal density on the capillary axis and maximal at the capillary wall. Magnetohydrodynamic simulations were performed for a range of prepulse parameters that defined different degrees of inhomogeneity of this initial profile. It was found that the plasma density distribution at the start of the main current pulse affects the cylindrical shock wave that takes place during the compression stage of a capillary discharge: A lower degree of radial inhomogeneity results in a steeper front of the shock wave. It is further shown that a steeper wave front results in a more concave electron density profile moments before the shock wave collapses on the capillary axis, when the EUV laser pulse presumably takes place, which may lead to a decrease in the rate of refraction losses. The proposed interpretation of the obtained numerical results correlates well with the available experimental data on the dependence of EUV laser pulse intensity and duration on the preliminary pulse amplitude.