Several recent studies of the occurrence of post-flutter limit cycle oscillations (LCO) of the F-16 have provided good support to the long-standing hypothesis that this phenomenon involves a nonlinear structural damping. The proposed mechanism for the appearance of nonlinearity in the damping are the nonlinear geometric effects that arise when the deformations become large enough to exceed the linear regime. In these investigations, a finite-element based reduced order modeling (ROM) framework with nonlinearity in damping has been developed. In this approach, the aircraft material is assumed to be viscoelastic with a dissipation tensor proportional to the elasticity tensor. The coefficient of proportionality, denoted as γ, represents the single tunable parameter of the model. With an appropriate calibration of this parameter, a good match between flight test LCO amplitudes and those predicted by the model has been obtained on average for each configuration tested. Fluctuations between predictions and flight test data have however been observed that suggest the presence of uncertainties, both aleatoric (e.g., variability from aircraft to aircraft) but also likely epistemic (i.e., unmodeled dynamics) ones. Accordingly, the focus of the present investigation is on developing a stochastic model of the F-16 aeroelastic response with uncertainty on the nonlinear damping properties and assessing the resulting band of uncertainty for some configurations analyzed in prior efforts.