The present work is a pioneering study on the contact mechanics including Pasternak foundation model. The context of this research is frictionless plane contact problem between a rigid punch and a functionally graded orthotropic layer lying on a Pasternak foundation in the limits of the linear elasticity theory. The layer is pressed by rigid cylindrical or flat punches that apply a concentrated force in the normal direction. The orthotropic material parameters are assumed to vary exponentially in the in-depth direction. Applying the Fourier integral transform technique and the boundary conditions of the problem, a singular integral equation is obtained, in which the contact stress and the contact width are unknowns. Using the Gauss–Chebyshev integration formula the singular integral equation is solved numerically. Effects of the Pasternak foundation parameters, material inhomogeneity, external load, punch radius or punch length on the contact stress, the contact width, the vertical displacements on the top and bottom surfaces of the layer, the subsurface and in-plane stresses are given.