A perturbative three-dimensional analysis is presented of Alfvén waves in a magnetic X-point configuration with a strong longitudinal guide field. The waves are assumed to propagate in the direction of the X-line, and both the plasma beta and equilibrium plasma current are taken to be zero. This provides a simple model of Alfvén wave propagation in the divertor region of tokamak plasmas. It is shown that the presence of the X-point places constraints on the structure of the leading-order (shear Alfvén) eigenfunctions. These eigenfunctions, and fast wave corrections to them, are determined explicitly for two cases. In the first of these the stream function for the shear Alfvén flow is azimuthally symmetric in the Xpoint plane and singular at the X-line; in the second case the stream function is largely confined to two quadrants in the X-point plane and is non-singular. For the latter scenario it is shown that coupling of the shear and fast waves is strongly localized to the vicinity of the separatrix.