The nonlinear behavior of the soil demands the direct method of the analysis of a soil-structure interaction (SSI) problem. In the direct method, the soil and structure systems are solved simultaneously using a finite element or finite difference method. However, this method is computationally expensive and demands high-computational resources in terms of storage and computational time. Further, reliability analysis of SSI problems needs repeated runs of the original finite element code for different values of uncertain parameters. This increases the computational cost significantly. In this paper, this issue is addressed systematically using reduced order models (ROMs). ROMs are successfully applied in various engineering fields, such as fluid dynamics, optimization, and structural dynamics. However, their applicability to SSI problems still needs to be addressed. The current work explores the suitability of a non-intrusive ROM for nonlinear SSI problems. A non-intrusive ROM does not require knowing the governing equations explicitly and thus can be used with any commercial solvers. First, the accuracy of the ROM is established for a beam on nonlinear Winkler foundation. Then, their efficiency is studied for reliability analysis. The estimates from the ROM are compared with direct Monte Carlo simulation, importance sampling, and the first-order reliability method. Numerical studies show that the ROM yields accurate results and efficiently estimates the reliability or failure probability. It is concluded from the numerical observations that the ROM is an efficient alternative tool for the reliability analysis of SSI problems.