In block-sparse-based compressed sensing, a block of indices is recovered from a non-adaptive random sample, which requires less computational time. Nonetheless, high-dimensional signals demand large storage spaces for sensing matrices in signal reconstruction. In this paper, a modified block sensing matrix is constructed from an initial dense submatrix. Its elements are drawn from an identically independent random Gaussian distribution. The full sensing matrix is not required in all intermediate computations. And, the required subset of sub-matrices can be generated at any stage of computation. It requires less storage space. The natural signal-like image does not exhibit any block-sparse property. In this paper, such signals are turned into blocks of sparse signals with suitable arrangements. The proposed sensing matrix provides efficient recovery of block-sparse signals using the proposed modified block orthogonal matching pursuit (MEBOMP) algorithm with proper adjustment. The results and analysis show the better performance of the proposed methods over other sensing matrices.