Natural disasters are unavoidable and can cause serious damage to bridges, which may lead to catastrophic losses, both human and economic. Therefore, the assessment of bridges exposed to these events is of paramount importance to identify possible mitigation needs. The objective of the present work is to present consistent tools that may allow us to obtain the failure probability of a masonry arch bridge under a flood event, leading to local scour. Surrogate models were implemented to ease the computational cost of the probabilistic analysis. Moreover, a stochastic parametric analysis based on the geotechnical properties of the soil components of masonry arch bridges located in Portugal was performed. The results show the failure mechanism of the masonry arch bridges when subjected to scour-induced settlements and the influence of soil density on the failure probability obtained for different flow discharge values and angles of attack. The presented methodology and derived fragility curves can be used to assess bridge performance under a flood event, thus providing useful information for bridge management and monitoring.