We study the extremal behaviour of some integer-valued bivariate time series. Assuming that the distributional behaviour of the innovations is the one introduced in Hüsler et al. (Methodol Comput Appl Probab 24:2373–2402, 2022), we establish asymptotics for the distribution of the bivariate normalized maxima of two max-BINAR models and of a BINMM model. Since the marginal distribution functions of these processes belong to Anderson’s class, we consider two different approaches. First, maintaining the Anderson’s setup, we establish limiting lower and upper bounds for the normalized double maximum. In a second step, considering the double maximum of the first \(k_n\) observations, where \(\{k_n\}\) is a non decreasing sequence of positive integers with an asymptotic geometric pattern, we obtain a well defined limit in distribution for the double maximum, which is a bivariate max-semistable distribution function (Pancheva, Theory Probab Appl, 679–705, 1992). In both cases the asymptotic independence of maxima is established.