Abstract This paper deals with two-sided matching with budget constraints where one side (firm or hospital) can make monetary transfers (offer wages) to the other (worker or doctor). In a standard model, while multiple doctors can be matched to a single hospital, a hospital has a maximum quota: the number of doctors assigned to a hospital cannot exceed a certain limit. In our model, a hospital instead has a fixed budget: the total amount of wages allocated by each hospital to doctors is constrained. With budget constraints, stable matchings may fail to exist and checking for the existence is hard. To deal with the nonexistence of stable matchings, we extend the “matching with contracts” model of Hat- field and Milgrom, so that it handles near-feasible matchings that exceed each budget of the hospitals by a certain amount. We then propose two novel mechanisms that efficiently return such a nearfeasible matching that is stable with respect to the actual amount of wages allocated by each hospital. In particular, by sacrificing strategy-proofness, our second mechanism achieves the best possible bound