Research on the instability of viscoelastic annular liquid jets in a radial electric field has been carried out. The analytical dimensionless dispersion relation between unstable growth rate and wave number is derived by linear stability analysis. The Oldroyd B model was used to describe the viscoelastic characteristics of the viscoelastic fluids. Considering that the para-sinuous mode has been found to be always dominant in the jet instability, the effects of various parameters on the instability of viscoelastic annular liquid jets are examined only in the para-sinuous mode. Nondimensionalized plots of the solutions exhibit the stabilizing or destabilizing influences of electric field effects and the physical properties of the liquid jets. Both temporal instability analysis and spatiotemporal instability analysis were conducted. The results show that the radial electric field has a dual impact on viscoelastic annular liquid jets in the temporal mode. Physical mechanisms for the instability are discussed in various possible limits. The effects of Weber number, elasticity number, and electrical Euler number for spatiotemporal instability analysis were checked. As the Weber number increases, the liquid jet is first in absolute instability and then in convective instability. However, the absolute value of the absolute growth rate at first decreases, and then increases with the increase of We, which is in accordance with temporal instability analysis. Comparisons of viscoelastic annular jets with viscoelastic planar liquid jets and cylindrical liquid jets were also carried out.