The nonfragile asynchronous control method is proposed for Takagi-Sugeno (T-S) fuzzy Markov jump systems (MJSs) with hybrid cyber-attacks and time-varying delays. Firstly, the Markov jump system model is established by utilizing the T-S fuzzy method under fuzzy rules. The hybrid cyber-attacks consisting of deception attacks and denial-of-service (DoS) attacks are considered in the communication network, and the dynamic event-triggered communication mechanism is adopted in the measurement channel to reduce unnecessary data transmission. Secondly, sufficient conditions for the stochastic stability of the closed-loop system are derived in terms of the Lyapunov-Krasovskii method and linear matrix inequality technique. The linear matrix inequality technique is used to solve the nonfragile asynchronous control gain matrices. Lastly, simulation examples verify the correctness and effectiveness of the proposed method.