We consider a family of multivariate distributions with heavy-tailed margins and the type I elliptical dependence structure. This class of risks is common in finance, insurance, environmental and biostatistic applications. We obtain the asymptotic tail risk probabilities and characterize the multivariate regular variation property. The results demonstrate how the rate of decay of probabilities on tail sets varies in tail sets and the covariance matrix of the elliptical copula. The theoretical results are well illustrated by typical examples and numerical simulations. A real data application shows its advantages in a more flexible dependence structure to characterize joint insurance losses.