We develop two novel couplings between arbitrary pure-jump Lévy processes to obtain upper bounds on the Wasserstein distance. The couplings are used to establish upper bounds on the rate of convergence of the Wasserstein distance on the path space for a wide class of Lévy processes attracted to a multidimensional stable law in the small-time regime. These bounds are shown to be rate-optimal for a wide class of processes in the domain of normal attraction. In fact, we show that the rate of convergence is polynomial for the domain of normal attraction and slower than any polynomial for the domain of non-normal attraction.