This paper investigates the asynchronous event-triggered state-feedback control for discrete-time nonlinear singular Markov jump systems subject to reachable set bounding and stochastic cyber-attacks. An event-triggered mechanism is introduced to regulate data transmission and save network resources. While this mechanism brings advantages, it also causes that the mode information of the system and the controller are not synchronized. On the other hand, network communication has brought convenience to signal transmission, and it has also resulted in the network channel being in an open and unreliable environment, which is vulnerable to cyber-attacks. In order to solve these problems, the secure asynchronous controller is designed to effectively resist the influence of attacks. In addition, by using the reachable set analysis approach, the constructed controller ensures that the estimation error is within the boundary of the ellipsoid, and the existence conditions of the required controller are given by solving the linear matrix inequalities. Finally, a simulation example is given to illustrate the effectiveness of the theoretical results.