In this letter, robust sparse signal recovery is considered in the presence of heavy-tailed impulsive noise. Two Bayesian approaches are developed where a Bayesian framework is constructed by utilizing the Laplace distribution to model the noise. By rewriting the noise-fitting term as a reweighted quadratic function which is optimized in the sparse signal space, the Type I Maximum A Posteriori (MAP) approach is proposed. Next, by exploiting the hierarchical structure of the sparse prior and the likelihood function, we develop the Type II Evidence Maximization approach optimized in the hyperparameter space. The numerical results verify the effectiveness of the proposed methods in the presence of impulsive noise.