The linear stability of a bilayer fiber spinning process is analyzed for axisymmetric disturbances under isothermal conditions. The co-extruded fiber consists of two concentric layers of rheologically stratified polymeric fluids. While the core fluid is an entangled polymer melt described using the eXtended Pom-Pom (XPP) model, the sheath layer is a low molecular weight unentangled polymer modeled as an Upper Convected Maxwell (UCM) fluid. The analysis indicates that, for fast extensional flows, such an arrangement enhances the critical draw ratio of the process over that found from the spinning of XPP fluid alone, thus delaying the onset of draw resonance and stabilizing the system. Further, the range of flow Deborah number for stable spinning is found to be much broader than that for the UCM fluid alone. For low to moderately elastic flows, the stability behavior is governed mainly by the rheology of the more-elastic core layer (XPP fluid), whereas for the highly elastic flows, the stability behavior is dominated by the less-elastic sheath layer (UCM fluid). The sensitivity of the stability diagram with respect to various spinning flow parameters, like the relative fraction of core and sheath layers, the rheological stratification of two fluids, and the spine-line dimensions, has also been examined in order to identify the region of enhanced stability such that the draw resonance is suppressed. © 2017 Elsevier Ltd