We propose a compressed sensing algorithm termed variance state propagation (VSP) for block-sparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. The VSP algorithm is developed under the Bayesian framework. A hierarchical Gaussian prior is introduced to depict the clustered patterns in the sparse signal. Markov random field (MRF) is introduced to characterize the state of the variances of the Gaussian priors. Such a hierarchical prior has the potential to encourage clustered patterns and suppress isolated coefficients whose patterns are different from their respective neighbors. The core idea of our algorithm is to iteratively update the variances in the prior Gaussian distribution. The message passing technique is employed in the design of the algorithm. For messages that are difficult to calculate, we correspondingly design reasonable methods to achieve approximate calculations. The hyperparameters can be updated within the iteration process. Simulation results demonstrate that the VSP algorithm is able to handle a variety of block-sparse signal recovery tasks and presents a significant advantage over the existing methods.