The objective of the work is first to extend some classic high cycle fatigue (HCF) criteria (as Crossland, Dang Van, Papadopoulos, ...) to take into account a sensitivity of the criteria to stress spatial variations occurring at length scale lg, and second to compare the performances of the extensions through numerical simulations of experimental fatigue tests. After an introduction of the basic criteria and their gradient based extensions proposed by Luu et al., we focus on the Crossland criterion to propose a more practical and simple expression taking into account the gradient of the stress amplitude and the maximum hydrostatic stress. The proposition is then tested and applied to different simple situations: 4-point bending and cantilever rotative bending. The relative errors between the exact solutions and the numerical simulations are estimated. Biaxial bending-torsion tests are also simulated to demonstrate the capabilities of the approach. The generalization of the approach to other multiaxial fatigue criteria is briefly shown through the case of Papadopoulos 2001 proposal. Finally, the present study develops a simple formulation of gradient multi-axial fatigue criteria extending the classical HCF criteria. In this work only stress gradient with a beneficial effect on fatigue have been considered.