Conductivity of tissues in the vicinity of the sources is especially influential on the MEG and EEG forward fields. Those tissues include white matter, whose conductivity is anisotropic because of its fiber structure. While white matter anisotropy can be measured thanks to Diffusion-Weighted MRI, it is rarely incorporated in MEG and EEG head models. Boundary Element Methods can only deal with piecewise constant conductivities, therefore ruling out white matter anisotropy that has a complex structure, and Finite Element Method have been developed to deal with anisotropic conductivity, but require very fine meshes, thus huge linear systems. The purpose of this paper is to extend the BEM framework to incorporate white matter anisotropy by treating anisotropic conductivity as a perturbation of an isotropic one.