The draw resonance instability in a concentric bicomponent coextrusion fiber spinning has been studied experimentally using a linear low density polyethylene (LLDPE) for the core and two low density polyethylenes (LDPE) for the skin-layer. These materials have been chosen to investigate a coextrusion flow in which each components have quite different extensional rheology. When they were spun individually, the LLDPE showed the draw resonance at a certain critical draw ratio whereas the LDPEs did not show the oscillatory instability. When the LLDPE and LDPE were spun together, the presence of the more viscoelastic LDPE layer delayed the onset of draw resonance of the LLDPE to a higher draw ratio. The degree of changes in the critical draw ratio was observed to be dependent on the rheological properties of the LDPEs. Although the experiments were conducted in a non-isothermal condition, various aspects of the experimental observations appeared to be in qualitative agreement with the predictions of isothermal linear theories. The present study is specifically for a coextrusion fiber spinning. Nevertheless, the observed trend of the instability should be also relevant to other coextrusion processes such as extrusion coating or film casting.