In this paper, we propose a new bivariate geometric model, derived by linking two univariate geometric distributions through a specific copula function, allowing for positive and negative correlations. Some properties of this joint distribution are presented and discussed, with particular reference to attainable correlations, conditional distributions, reliability concepts, and parameter estimation.AMonteCarlo simulation study empirically evaluates and compares the performance of the proposed estimators in terms of bias and standard error. Finally, in order to demonstrate its usefulness, the model is applied to a real data set.