The propagation behavior of Love waves in a functionally graded material layered half-space with initial stress is taken into account. The Wentzel-Kramers-Brillouin (WKB) asymptotic technique is adopted for the theoretical derivations. The analytical solutions are obtained for the dispersion relations and the distributions of mechanical displacement and stress along thickness direction in the layered structure. Firstly, these solutions are used to study effects of the initial stress on the dispersion relations and phase velocities, then influences of the initial stress on the distributions of mechanical displacement and shear stresses along thickness direction are discussed in detail. Numerical results obtained indicate that the phase velocity of Love wave increases with the increase of the magnitude of the initial tensile stress, while decreases with the increase of the magnitude of the initial compression stress. The effects on the dispersion relations of the Love wave propagation are negligible as the magnitudes of the initial stress are less than 100MPa. Some other results are shown for distributions of field quantities along thickness direction. The results obtained are not only meaningful for the design of functionally graded structures with high performance but also effective for the evaluation of residual stress distribution in the layered structures.