A general price process represented by a two-component Markov process is considered. Its first component is interpreted as a price process and the second one as an index process modulating the price component. American type options with pay-off functions, which admit power type upper bounds, are studied. Both the transition characteristics of the price processes and the pay- off functions are assumed to depend on a perturbation parameter d= 0 and to converge to the corresponding limit characteristics as d¿ 0. In the first part of the paper, asymptotically uniform skeleton approximations connecting reward functionals for continuous and discrete time models are given. In the second part of the paper, these skeleton approximations are used for getting results about the convergence of reward functionals for American type options for perturbed price processes in discrete and continuous time. Examples related to modulated exponential price processes with independent increments are given.