Structural reliability analysis provides practitioners with tools to estimate the probability of failure of various engineering systems. Failure is characterized by a so-called limit-state function that takes as input a set of uncertain variables describing the system. Due to the complexity of engineering systems, multiple limit-state functions are generally needed. The problem is known as system reliability, as opposed to component reliability, for which only a single limit-state is considered. Surrogate-based solution schemes have shown to be the most efficient methods when used in an active learning scheme. Research efforts have mainly been devoted to component reliability analysis. Extensions or adaptations to system reliability have been proposed but they lack of efficiency. In this work, we propose an active learning scheme for solving system reliability problems in an arbitrary configuration while accounting for the difference in evaluation costs of the various limit-states. We use Sobol’ sensitivity analysis and clustering to identify the relevant limit-state functions to update at each iteration. We then formulate a discrete optimization problem that allows us to account for the computational budget constraints and each limit-state evaluation cost. The proposed method is validated on an analytical example.