In this work, we consider the problem of stability of the bivariate classical risk model. In particular, we are interested in the stability bounds of the classical risk model established by Benouaret and Aïssani (2010) using the approach based on the strong stability method developed for general Markov chains. Based on numerical and graphical results, we study the stability bounds of the ruin probability considering different heavy-tailed claim distributions, and we show the impact of large claims on the stability bound of a bivariate classical risk model.