Electrokinetic flows involving fluids with non-uniform conductivity can trigger instabilities known as electrokinetic instabilities (EKI). Prior studies have established the theoretical framework of EKI in Newtonian fluids, where the electric Rayleigh number is identified to govern the onset of instabilities. We propose in this work a theoretical formulation of the electric Rayleigh number for EKI in viscoelastic fluids with conductivity gradients. Our scaling analysis in terms of the Oldroyd-B model predicts that fluid viscoelasticity suppresses EKI through the combined viscous and elastic effects. We also conduct experiments to investigate the effects of fluid viscoelasticity and viscoelasticity gradients on EKI in hyaluronic acid polymer solutions. Our observations verify the theoretical analysis and demonstrate that the presence of viscoelasticity gradients causes a further suppression of EKI with the extent strongly dependent on the direction of rheology gradients. Furthermore, we analyze the perturbation energy, spatiotemporal evolution, and power spectra to understand and compare the EKI dynamics among different fluid combinations.