In the present work, the stability of a viscoelastic fluid flow is studied by Linear Stability Theory (LST), and some results are verified by Direct Numerical Simulation (DNS). The investigation considers the fluid flow between two parallel plates, modelled by the Giesekus constitutive equation. The results show the influence of the anisotropic tensorial correction parameter $\alpha_G$ on this model, showing a stabilizing influence for two-dimensional disturbances for small values of $\alpha_G$. However, as $\alpha_G$ increases, a reduction in the critical Reynolds number values is observed, possibly hastening the transition to turbulence. Low values for $\alpha_G$ for three-dimensional disturbances cause more significant variations for the critical Reynolds number. This variation decreases as the value of this parameter increases. The results also show that low values of $\alpha_G$ increase the instability of three-dimensional disturbances and confirm that Squire's theorem is not valid for this model. As for the two-dimensional disturbances, the anisotropic term on the Giesekus model lowers the critical Reynolds number for higher quantities of polymer viscosity in the mixture and high values for the Weissenberg number.