We study theoretically the influence of viscoelasticity on the steady cone-jet mode of electrospray for small stress relaxation times. For this purpose, we numerically integrate the leaky-dielectric model together with the Oldroyd-B constitutive relationship and calculate both the base flow and linear eigenmodes characterizing its stability as a function of the governing parameters. We describe the effect of the polymeric stresses on both the cone-jet mode and the minimum flow rate stability limit. There are considerable differences between the Newtonian and viscoelastic electrospray realizations even for relatively small stress relaxation times due to the intense extensional deformation suffered by the fluid particles in the cone-jet transition region The axial polymeric stress shrinks the liquid meniscus and stabilizes it by pushing the fluid particle in the cone-to-jet transition region.