In this paper, we study a multidimensional risk model with a common renewal process and in the presence of a constant interest force. The claim sizes are independent and identically distributed random vectors, with the distribution of dependent components belonging to the class of multivariate subexponential distributions. We establish locally uniform asymptotic estimations for the entrance probability of the discounted aggregate claims into some rare sets, and further derive asymptotic estimations uniformly over all the time horizons. Furthermore, we present some distribution examples that belong to these multivariate heavy-tailed distribution classes, which are not restricted only to the case of multivariate regular variation.