Recently, a new distribution with bounded support called unit Gompertz has been derived by taking an exponential transformation from the parent Gompertz distribution. This distribution has right-skewed (unimodal) and reversed-J shaped density. Moreover, the hazard rate has constant, increasing, bathtub and upside-down bathtub. In this paper, the bivariate extension for this new distribution is introduced and its properties are discussed in detail. The new bivariate model is of the Marshall–Olkin type. The estimation problem for the model's unknown parameters has been considered using MLE and Bayesian estimation methods; fortunately, the Bayes estimators are theoretically obtained in explicit forms. Furthermore, the Bayesian estimators are computed using MCMC method. Two real data sets have been applied to the bivariate unit Gompertz distribution. Some simulations are carried out to see the performances of the estimators. Absolutely continuous bivariate versions of this new distribution are obtained and some of its properties are also discussed.