The development of combination therapies has become commonplace because potential synergistic benefits are expected for resistant patients of single-agent treatment. In phase I clinical trials, the underlying premise is toxicity increases monotonically with increasing dose levels. This assumption cannot be applied in drug combination trials, however, as there are complex drug-drug interactions. Although many parametric model-based designs have been developed, strong assumptions may be inappropriate owing to little information available about dose-toxicity relationships. No standard solution for finding a maximum tolerated dose combination has been established. With these considerations, we propose a Bayesian optimization design for identifying a single maximum tolerated dose combination. Our proposed design utilizing Bayesian optimization guides the next dose by a balance of information between exploration and exploitation on the nonparametrically estimated dose-toxicity function, thereby allowing us to reach a global optimum with fewer evaluations. We evaluate the proposed design by comparing it with a Bayesian optimal interval design and with the partial-ordering continual reassessment method. The simulation results suggest that the proposed design works well in terms of correct selection probabilities and dose allocations. The proposed design has high potential as a powerful tool for use in finding a maximum tolerated dose combination.