The univariate Chen distribution has a number of appealing characteristics. It behaves similarly to Weibull, gamma, and generalized exponential distributions with two parameters. In this paper, we consider the bivariate mixture models starting with two independent univariate Chen models. Several useful structural properties of such a mixture model are discussed. Inferential aspects under the classical method are considered to estimate the model parameters. For illustrative purposes, a well- known motor data is re-analyzed to exhibit the flexibility of this proposed bivariate mixture model