In this study, the plane contact problem for a rigid cylindrical punch and a functionally graded bilayer is considered. The layers have different thicknesses and elastic constants. The normal and tangential forces are applied to the upper layer with a rigid cylindrical punch, and the lower layer is fully bonded to a rigid substrate. Poisson's ratios are taken as constant, and elasticity moduli are assumed to vary exponentially through the thickness of the layers. With the use of Fourier integral transform, the plane contact problem is reduced to a singular integral equation in which the unknowns are the contact pressure and the contact width. The singular integral equation is solved numerically using Gauss-Jacobi integration formula. The effect of several geometrical and physical parameters such as the material inhomogeneity, the friction coefficient, the layers' height, the mismatch in the material properties at the interface, and the contact width on the contact stress and in-plane stress are investigated in detail.