We study both the pure anisotropic and the anisotropically diluted transverse Ising model at zero temperature on the square lattice. In the former the couplings along each Cartesian direction are allowed to be different, whereas in the latter it is the bond concentration which may be different along the Cartesian directions. By means of a position space renormalisation group scheme, we find that in both cases the isotropic systems are stable with respect to anisotropy perturbations. We also discuss our results in view of r recently proposed criterion for the stability of pure system fixed points with respect to correlated dilution.