Lévy copulas are an important tool which can be used to build dependent Lévy processes. In a classical setting, they have been used to model financial applications. In a Bayesian framework they have been employed to introduce dependent nonparametric priors which allow to model heterogeneous data. This paper focuses on introducing a new class of Lévy copulas based on a class of subordinators recently appeared in the literature, called \textit{Compound Random Measures}. The well-known Clayton Lévy copula is a special case of this new class. Furthermore, we provide some novel results about the underlying vector of subordinators such as a series representation and relevant moments. The article concludes with an application to a Danish fire dataset.