In this paper, we propose a SAT-based method for finding attractors of bounded size in asynchronous automata networks. The automata network is a multi-valued mathematical model which has been studied for the quali- tative modeling of biological regulatory networks. An attractor is a minimal set of states in automata networks that cannot be escaped and thus loops indefinitely. Attractors are crucial to validate the initial design of a biological model and predict possible asymptotic behaviors, e.g., how cells may result through maturation in differentiated cell types. Developing an efficient computational method to find attractors is thus an important research topic. Our contribution is a translation of the problem of finding attractors of automata networks into a sequence of propositional satisfiability (SAT) problems. We also propose to add two optional constraints to improve the computation time of attractors. Experiments are carried out using 30 automata networks, 8 coming from real biological case studies and 22 crafted ones with controlled attractor size. The experimental results show that our method scales better than the state-of-the-art ASP method when the size of the attractors increases.