We investigate the effect of isotropic velocity-dependent potentials on the bound state energy eigenvalues for the first time for any quantum states of the Coulomb and harmonic oscillator potentials within the framework of the asymptotic iteration method.When the velocity-dependent term is selected as a constant parameter po,we present that the energy eigenvalues can be obtained analytically for both Coulomb and harmonic oscillator potentials.However, when the velocity-dependent term is considered as a harmonic oscillator type por2, taking the velocity-dependent term as a perturbation, we present how to obtain the energy eigenvalues of the Coulomb and harmonic oscillator potentials for any n and e quantum states by using perturbation expansion and numerical calculations in the asymptotic iteration method procedure.