We consider wormhole geometries subject to a gravitational action consisting of nonlinear powers of the Ricci scalar. Specifically, wormhole throats are studied in the case where Einstein gravity is supplemented with a Ricci-squared and inverse Ricci term. In this modified theory, it is found that static wormhole throats respecting the weak energy condition can exist. The analysis is done locally in the vicinity of the throat, which eliminates certain restrictions on the models introduced by considering the global topology.