Passive vibration isolators are widely used in multiple engineering applications to reduce resonance peaks or to mitigate transmissibility in the presence of internal or external sources of dynamic excitation. The design of a linear passive isolator involves multiple trade-offs. In the literature, different design configurations with nonlinearities have been investigated to limit some of these trade-offs. These include designs with quasi-zero stiffness (QZS) or high-static-low-dynamic stiffness (HSLDS) characteristics. This study investigates three viscoelastic models that incorporate stiffness nonlinearity along the non-isolating axes in order to exhibit more control over the dynamic response of the isolated system and possibly mitigate some of the design trade-offs. The dynamic response of these three models is compared to an existing HSLDS model in the literature. The three models investigated in this study are as follows: Kelvin-Voigt (or Voigt), Zener, and Generalized Maxwell (or Maxwell Ladder). These three models have been commonly used in the literature for vibration analysis of passive isolators. Two methods have been used for analysis, namely the Harmonic Balance Method (HBM) and explicit numerical integration. Test results from a previous study have been used for model characterization of all the models. It is observed that the modified Kelvin-Voigt model is analogous to the HSLDS model from the literature. For the isolator parameters used in this study, it is observed that the Kelvin-Voigt model with stiffness nonlinearity is able to exhibit characteristics similar to the HSLDS design, this includes the jump phenomenon as well as the hardening behavior. In general, all three models demonstrate that stiffness nonlinearity results in a reduction in peak transmissibility as well as an enhancement of the isolation bandwidth. The findings of this study could be useful in the design of passive isolation systems for products with significantly different multi-axial requirements with various design constraints.