The elastic stability of a wound coil comprising a central core and many layers of sheet metal is modeled and analyzed. A common failure mode resulting from unfavorable internal stresses—called v-buckling—is characterized by a section of the core buckling inward, possibly with several nearby sheet metal layers. In the present study, the core is modeled as a thin cylinder that is subjected to (i) the uniform external pressure generated by the coil's wound-in stresses and (ii) a nonuniform elastic foundation around its circumference that represents core-coil contact or loss thereof. The model and an iterative numerical technique are used to predict the critical winding pressure along the core-coil interface and the core's ensuing buckled shape. The role of geometric imperfection in the core, and the sensitivity of the buckling pressure to such initial defects, are also examined. Critical imperfection wavenumbers that facilitate the onset of significant deformations are identified with a view toward applying the results to improve quality and core inspection procedures. The predicted buckling pressure and the maximum radial stress developed in the coil, as based on a nonlinear stress model, are together used to determine factors of safety against core buckling over a range of manufacturing process parameters. Three case studies evaluate sensitivity with respect to process tension, core radius, and core thickness. The results are intended to guide the development of solutions to control the stability and quality of coils in sheet metal manufacturing.