Software testing activities are critical for the development of high-quality software systems. In this context, high-quality models that accurately represent the system under test are an essential tool. To improve their quality, the inference of these models has been addressed as a multi-objective optimization problem in the literature. In this work, we propose a method based on the Multi-Objective Variable Neighborhood Descent (MO-VND) scheme to solve the problem of inferring behavioral models of software systems built on user-reported bugs. To configure the MO-VND we introduce five different neighborhood structures. The performance of the method is evaluated on a benchmark of real-world instances. The results show that the order in which the neighborhoods are explored greatly affects the performance of the MO-VND method. We compare the proposed method with three well-known algorithms that have already been studied in the literature for this problem: NSGA-II, NSGA-III, and MOEA/D. The results show that, although the proposed MO-VND method is capable of finding high-quality models, there is still room for improvement. In particular, the method might benefit from strategies for an efficient evaluation of neighbor solutions or the identification of promising solutions within the proposed neighborhoods.