High-frequency asymptotic methods, based on solving the eikonal equation, are widely used for many seismic processing and imaging applications. For an attenuating medium, the eikonal equation becomes complex. The real part of the resulting complex traveltime describes the wave propagation while the imaginary part is associated with attenuation effects. During the past decades, several methods have been proposed to numerically solve the eikonal equation for non-attenuating media. However, solving a complex eikonal equation numerically involves several complications. We propose a fast sweeping algorithm to approximate the solution of the acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis (VTI). We implement a perturbation method to derive the governing equations for the zeroth- and the first-order coefficients of the traveltime expansion. Numerical tests show that the proposed scheme is applicable to VTI media with weak attenuation.