We numerically investigate inertial flows of viscoelastic fluids within a three-dimensional cross-slot geometry of square cross-section. Our study focuses on the inertial instability that occurs above a critical Reynolds number (Re) resulting in a transition to a steady flow asymmetry. We investigate numerically the effects of elasticity upon its characteristics by employing the upper-convected Maxwell (UCM), the Oldroyd-B and the modified Chilcott Rallison finitely extensible nonlinear elastic (FENE MCR) models. In so doing, we show that the UCM and the Oldroyd-B model results are restricted to very low nominal Weissenberg numbers at non-negligible Reynolds numbers, due to a significant increase in the strain rate at the stagnation point caused by inertia. The resulting steady-asymmetric flow at these critical conditions gives rise to the formation of a single axially-aligned spiral vortex, which is formed along the outlet channels of the geometry in good agreement with experimental observations. Below these critical conditions the flow remains steady-symmetric, varying from a nearly two-dimensional flow at very low Re to a more complex three-dimensional flow at higher Re. In a recent publication [Burshtein et al. [Phys. Rev. X, 7, 041039, (2017)]], we demonstrated experimentally, accompanied by limited complementary numerical simulations, the impact of elasticity upon the critical conditions for which the instability develops and the behaviour of the subsequent growth of vorticity of the single spiral vortex. Our results here show how different viscoelastic models influence the instability in a 3D cross-slot and demonstrate an interesting behaviour of the first normal-stress difference, providing an additional insight on the potential mechanisms which are responsible for the suppression of the spiral-vortex. We also elucidate the role played by solvent-to-total viscosity ratio and the extensibility parameter in the FENE MCR model on the instability.