The linear stability analysis is carried out for the straight jet of a polymer melt in an electrospinning process. The stability of axisymmetric disturbances is examined in order to comprehend the onset of fiber morphology with diametric variations or bead formation along the fiber under non-isothermal electrospinning conditions. As the polymeric fluid (polylactic acid melt) is a low conductivity fluid with unentangled polymer molecules, the viscoelasticity is described using the non-linear Theological Giesekus constitutive model assuming very small axial conduction current. The non-periodic axisymmetric disturbances are imposed on the non-uniform radius jet, obtained as the solution of the 1-D slender filament governing equations and the eigenspectrum for the disturbance growth rate is constructed under the realistic melt electrospinning conditions. The growth rate corresponding to the leading mode in the eigenspectrum is found to increase with increasing surface tension forces and decrease with the enhanced external electric field. Thus, the leading growth rate behavior suggests that the classical Rayleigh-Plateau instability dominates over the conducting mode of instability for melt electrospinning. Further, the role of non-isothermal conditions in the stability behavior is examined. The convective heat transfer from electrified jet to the cooling ambiance leads to thicker fibers with greater stability to axisymmetric disturbances. The stabilizing effect of heat transfer is attributed mainly to the temperature sensitive fluid rheology. In particular, the enhancement in polymer viscosity in the jet propagation direction is responsible for the build up of stabilizing viscoelastic stress. The fluid elasticity, denoted by the flow Deborah number, also tends to stabilize the electrified jet as temperature drop along the flow increases the relaxation time of the polymer chains leading to high polymeric stress associated with the stretched chains. As crystallization of polymeric chains under non-isothermal condition is not considered, the analysis holds for either amorphous polymers or polymers with slow crystallization kinetics compared to the short residence time during the spinning flow.