We consider the problem of recovering the common support of a set of k-sparse signals {x_i}L_i=1 from noisy linear underdetermined measurements of the form {Φx_i + w_i}L_i=1 where Φ ϵ R^m×N (m <; N) is the sensing matrix and wi is the additive noise. We employ a Bayesian setup where we impose a Gaussian prior with zero mean and a common diagonal covariance matrix Γ across all xi, and formulate the support recovery problem as one of covariance estimation. We develop an algorithm to find the approximate maximum-likelihood estimate of Γ using a modified reweighted minimization procedure. Empirically, we find that the proposed algorithm succeeds in exactly recovering the common support with high probability in the k <; m regime with L of the order of m and in the k ≥ m regime with larger L. The key advantage of the proposed algorithm is that its complexity is independent of L, unlike existing sparse support recovery algorithms.