This paper addresses the problem of Bayesian Block Sparse Modeling when coefficients within the blocks are correlated. In contrast to the current hierarchical methods which do not exploit correlation structure within the blocks, we propose a three level hierarchical estimation framework. It employs heavy-tailed priors for block sparse modeling and variational inference for Bayesian estimation. This paper also describes the relationship between proposed framework and some of the existing Block Sparse Bayesian Learning (SBL) methods and show that these SBL methods can be viewed as its special cases. Extensive experimental results for synthetic signals are provided, demonstrating the superior performance of the proposed framework in terms of failure rate, relative reconstruction error, to name a few. We also demonstrate the applicability of this framework in telemonitoring of Fetal Electrocardiogram. © 2018 IEEE.