We study the nonlinear fluid-structure interaction of an elastically supported rigid circular cylinder in a laminar flow. Periodic shedding of counter-rotating vortices from either side of the cylinder results in vortex-induced vibration of the cylinder. We demonstrate the passive suppression of the limit cycle oscillation (LCO) of the cylinder with the use of an essentially nonlinear element, the nonlinear energy sink (NES). The computational study is performed at a Reynolds number (Re) of 100; Re is defined based on the cylinder diameter and inlet velocity. The variational multiscale residual-based stabilized finite-element method is used to compute approximate solutions of the incompressible Navier-Stokes equations. The NES is comprised of a small mass, an essentially nonlinear spring, and a linear damper. With appropriate values for the NES parameters, the coupled system of flow-cylinder-NES exhibits resonant interactions, resulting in targeted energy transfer (TET) from the flow via the cylinder to the NES, where the energy is dissipated by the linear damper. The NES interacts with the fluid via the cylinder by altering the phase relation between the lift force and the cylinder displacement; this brings about significant reduction in the LCO amplitude of the cylinder for several set of values of the NES parameters.