The present paper deals with the flow of an incompressible couple stress fluid past a porous spheroidal shell which is made up of two confocal prolate spheroids S0 and S1 where S0 is within S1. The spheroid S0 is taken to be a solid and the annular region between S0 and S1 is porous, with the boundary of S1 being pervious. The flow outside S1 is governed by the linearized version of Stokes’ couple stress fluid flow equations and that within the porous region is governed by the classical Darcy’s law. The resulting equations are then solved analytically for the velocity and pressure fields and drag experienced by the body is obtained. The variation of drag with the different parameters like the material and the geometric is studied numerically and the results are presented through graphs. Stream lines are also plotted to understand the flow pattern.