Abstract Transverse vibrations of an inhomogeneous circular thin plate are studied. The plates, which geometric and physical parameters slightly differ from constant and depend only on the radial coordinate, are analyzed. After separation of variables the obtained homogeneous ordinary differential equations together with homogeneous boundary conditions form a regularly perturbed boundary eigenvalue problem. For frequencies of free vibrations of a plate, which thickness and/or Young’s modulus nonlinearly depend on the radial coordinate asymptotic formulas are obtained by means of the perturbation method. The effect of the small perturbation parameter on behavior of frequencies is analyzed under special conservation conditions: i) for a plate, the mass of which is fixed, if the thickness is variable, and ii) for a plate with the fixed average sti_ness, if Young’s modulus is variable. Asymptotic results for the lower vibration frequencies well agree with the results of finite element analysis with COMSOL Multiphysics 5.4.