During the last decades, reweighted procedures have shown high efficiency in computational imaging. They aim to handle non-convex composite penalization functions by iteratively solving multiple approximated subproblems. Although the asymptotic behaviour of these methods has recently been investigated in several works, they all necessitate the sub-problems to be solved accurately, which can be sub-optimal in practice. In this work we present a reweighted forward-backward algorithm designed to handle non-convex composite functions. Unlike existing convergence studies in the literature, the weighting procedure is directly included within the iterations, avoiding the need for solving any sub-problem. We show that the obtained reweighted forward-backward algorithm converges to a critical point of the initial objective function. We illustrate the good behaviour of the proposed approach on a Fourier imaging example borrowed to radio-astronomical imaging.