The radial basis integral equation (RBIE) method is derived for the first time to solve potential problems involving material anisotropy. The coefficients of the anisotropic conductivity require the gradient term to be modified accordingly when deriving the boundary integral equation so that the flux expression can be properly accounted. Analyses of the behavior of the anisotropic fundamental solution and its spatial gradients showed that their variations along the subdomain boundaries may be large and they increase as the diagonal components of the material anisotropy become larger. The accuracy of the anisotropic RBIE was found to depend primarily on the accuracy of the influence coefficients evaluations and this precedes the number of nodes used. Root mean squared errors of less than 10-4 can be obtained if evaluations of the influence coefficients are sufficiently accurate. An alternative formulation of the anisotropic RBIE was derived. The levels of accuracy obtained were not significantly different from the standard formulation.