Magnetohydrodynamic (MHD) flow refers to the study of behavior of electrically conductive fluid flow when external magnetic field is applied. It has numerous uses across various branches of engineering and industrial applications. It also has prominent application in biomedical field. The presence of an object in MHD flow field has prime importance for above applications. Accordingly, this work presents an attempt to apply the immersed boundary method (IBM) using fractional step based finite volume scheme to capture the interplay of both hydrodynamic and magnetic field past a circular and square cylinder within a channel. Numerical simulations are performed using a two-dimensional computational framework to analyze the unsteady Navier–Stokes equations in association with Lorentz force. Role of magnetic field in streamwise, transverse and oscillating directions on MHD flow past circular and square cylinder are studied. The findings indicate that as magnitude of magnetic field increases, the length of recirculation zone decreases. For the circular cylinder, the critical Hartmann numbers are found to be 5 and 7 in the streamwise and transverse directions, respectively, while for the square cylinder, they are 10 and 13 at Reynolds number of 40. In addition, impact of magnetic field, when circular shaped cylinder is oscillating, is also studied and noted that Hartmann number of 6 is required to suppress the recirculation zone completely at Reynolds number of 40.