In this paper, we investigate the pricing of variance swaps under a Markovian regime-switching extension of the Schöbel-Zhu-Hull-White hybrid model. The parameters of this model, including the mean-reversion levels and the volatility rates of both stochastic interest rate and volatility, switch over time according to a continuous-time, finite-state, observable Markov chain. By utilizing techniques of measure changes, we separate the interest rate risk from the volatility risk. The prices of variance swaps and related fair strike values are represented in integral forms. We illustrate the practical implementation of the model by providing a numerical analysis in a two-state Markov chain case, which shows that the effect of both stochastic interest rate and regime-switching is significant in the pricing of variance swaps.