We study the relaxation dynamics of a coarse-grained polymer chain at different degrees of stretching by both analytical means and numerical simulations. The macromolecule is modeled as a string of beads, connected by anharmonic springs, subject to a tensile force applied at the end monomer of the chain while the other end is fixed at the origin of coordinates. The impact of bond nonlinearity on the relaxation dynamics of the polymer at different degrees of stretching is treated analytically within the Gaussian self-consistent (GSC) approach and then compared to simulation results derived from two different methods: Monte Carlo (MC) and Molecular Dynamics (MD). At low and medium degrees of chain elongation we find good agreement between GSC predictions and the MC simulations. However, for strongly stretched chains, the MD method, which takes into account inertial effects, reveals two important aspects of the nonlinear interaction between monomers: (i) a coupling and energy transfer between the damped, oscillatory normal modes of the chain and (ii) the appearance of nonvanishing contributions of a continuum of frequencies around the characteristic modes in the power spectrum of the normal mode correlation functions.