The free vibration (FV) and dynamic stability (DS) analyzes are presented for functionally graded viscoelastic plates (FGVPs) under compressive load and resting on elastic foundations (EFs). Winkler elastic foundation models and Pasternak elastic foundation models are used as elastic foundations. The basic equations of FGVPs interacting with EFs are derived using the concepts of Boltzmann and Volterra. An analytical method for studying the DS and FV of FGVPs interacting with EFs is developed using the integro-differential equations. To solve the current problem, Galerkin and Laplace methods are used. A technique for the analysis of DS and FV of FGVPs on the EFs is developed. To confirm the proposed formulation, the results are compared with other available solutions. Finally, the influences of EFs, volume fractions and rheological constants on the critical times and frequencies depending on the geometrical characteristics and loading parameters are examined.