This paper proposes a new algorithm for total variation (TV) image deconvolution under the assumptions of linear observations and additive white Gaussian noise. By adopting a Bayesian point of view, the regularization parameter, modeled with a Jeffreys' prior, is integrated out. Thus, the resulting crietrion adapts itself to the data and the critical issue of selecting the regularization parameter is sidestepped. To implement the resulting criterion, we propose a majorization-minimizationapproach, which consists in replacing a difficult optimization problem with a sequence of simpler ones. The computational complexity of the proposed algorithm is O(N) for finite support convolutional kernels. The results are competitive with recent state-of-the-art methods.