We report an extensive nonequilibrium molecular dynamics investigation of the thermal conductivity of water using two of the most accurate rigid nonpolarizable empirical models available, SPC/E and TIP4P/2005. Our study covers liquid and supercritical states. Both models predict the anomalous increase of the thermal conductivity with temperature and the thermal conductivity maximum, hence confirming their ability to reproduce the complex anomalous behaviour of water. The performance of the models strongly depends on the thermodynamic state investigated, and best agreement with experiment is obtained for states close to the liquid coexistence line and at high densities and temperatures. Considering the simplicity of these two models the overall agreement with experiments is remarkable. Our results show that explicit polarizability and molecular flexibility are not needed to reproduce the anomalous heat conduction of water.