The design of passive vibration isolators involves multiple trade-offs. Design configurations with nonlinearities are commonly investigated to mitigate trade-offs, including designs with quasi-zero stiffness (QZS) and high-static-low-dynamic stiffness (HSLDS). This study investigates three viscoelastic models with stiffness nonlinearity along the non-isolating axes to control the dynamic response. These three models are: Kelvin-Voigt, Zener, and Generalized Maxwell. The Harmonic Balance Method (HBM) and explicit numerical integration are used for analysis and test results from an existing study have been used for model characterization. The modified Kelvin-Voigt model has been independently developed and is seen to be analogous to the HSLDS model in the literature, including the jump phenomenon as well as the hardening behavior. The introduction of stiffness nonlinearity is seen to result in a reduction in peak transmissibility as well as an enhancement of the isolation bandwidth, a behavior exhibited by all three models.