In magnetic resonance electrical impedance tomography (MREIT) where we try to reconstruct conductivity images with high spatial resolution, we should generate at least two non-parallel internal current density distributions by sequentially injecting multiple independent currents. However, when the object has a relatively very low conductivity region, the direction of current density due to any injection current becomes tangential to the boundary of the region. This kind of regions may include bones, lungs, and air-filled stomach. Hence, the previous MREIT reconstruction algorithms fail to provide accurate conductivity reconstruction near boundaries of insulating regions since this requirement is not satisfied. To deal with this difficulty, we propose local harmonic Bz-algorithm which find conductivity values in the separated problematic regions.