In this article, the asynchronous filtering scheme for Markovian jump systems (MJSs) subjected to time-varying delays and infinite distributed delays, is investigated based on the homogeneous polynomial approach. First, sufficient conditions are proposed to ensure that the filtering error system is exponentially stabile in mean square and satisfies a given performance index simultaneously. Second, the asynchronous filter is synthesized for MJSs with certain transition probability. Moreover, one so-called homogeneous polynomial method is developed to design the asynchronous filter in the case of uncertain transition probabilities. During the analysis, a homogeneous polynomial matrix is introduced in the parameter-dependent Lyapunov function that can reduce the conservatism of the results. Finally, a numerical example is given to support the feasibility and effectiveness of the proposed theory.