In this paper, we investigate the effect of magnetic field on two-dimensional flow of a viscous, incompressible fluid through Composite porous channel using non-primitive boundary element method (BEM). We consider a rectangular channel consisting of two packings that are filled with fully saturated porous medium. It is assumed that both the porous regions are homogenous and isotropic with different permeabilities. Brinkman equation governs the fluid flow through porous media. We analyze the effect of Hartman number, stress-jump coefficient, Darcy number, thickness parameter, electrical conductivity ratio, and viscosity ratio on fluid mechanics. We present the effect of stress-jump coefficients on the interfacial velocity of the fluid against the thickness parameter and observe that the interfacial velocity increases with increasing stress-jump coefficients. We notice that for a fixed value of thickness parameter, the magnitude of vorticity (at lower and upper walls) increases with increasing Darcy number. Moreover, we observe that the magnitude of vorticity at the lower wall decreases and increases at the upper wall with increasing thickness parameter. We compute the Brinkman layer thickness near the interface of the Composite porous channel in terms of several flow parameters and observe that the Brinkman layer thickness is strongly depend on the Hartman number, Darcy number, viscosity ratios, and stress-jump coefficient, respectively.