The Conway–Maxwell Poisson (CMP) distribution is a popular model for analyzing data that exhibit under or over dispersion. In this article, we construct bivariate CMP distributions with given marginal CMP distributions and range of correlation coefficient over (− 1, 1) based on the Sarmanov family of bivariate distributions. One of the constructions is based on a general method for weighted distributions. The dependence property is examined. Parameter estimation, tests of independence and adequacy of model and a Monte Carlo power study are discussed. A real data set is used to exemplify its usefulness with comparison to other bivariate models.