The discrete element method (DEM) has been extensively utilized to investigate the mechanical properties of granules, particularly their microscopic behavior, overcoming limitations in field tests such as cost, time consumption, and soil condition restrictions. To ensure the development of reliable DEM simulations, proper contact model selection and parameter calibration are essential. In this research, a DEM parameter calibration method that could represent the nonlinear relationship between clayey soil pressure and sinkage at different moisture contents was proposed. Firstly, the sinking modulus K and the soil deformation exponent n were identified to reflect the nonlinear pressure–sinkage relationship. Then, sensitive DEM parameters on the soli pressure–sinkage relationship were investigated and calibrated, and the effect of moisture content on them was explored. Finally, the transfer of soil internal stress during subsidence was analyzed using the constructed discrete element model. The average error of the sinking modulus K and the soil deformation exponent n between the DEM and the experimental result at four moisture contents were 4.7% and 4.9%, respectively. The relative error of soil internal stress between simulation and experiment was 6.7%, 4.4%, and 9.7% at depths of 50 mm, 100 mm, and 150 mm, respectively. The soil particle trajectory, soil internal stress distribution, and variations during plate pressure–sinkage progress were analyzed by the constructed DEM model. The results demonstrated good agreement with theoretical models and experimental findings. The proposed clayey soil DEM modeling process that considers the pressure–sinkage nonlinear relationship at different moisture contents can be applied in machine-soil research.