In recent shallow water experiments, the temporal and spatial focusing properties of time-reversing arrays (TRAs) were shown to be robust in a reciprocal medium and useful for underwater applications. The presence of oceanic currents in coastal environments leads to nonreciprocal acoustic propagation. In this case, time-reversal invariance is modified because the propagation speed inhomogeneity depends on the direction of acoustic propagation. Therefore, similarly to phase coherent reciprocal transmissions, a TRA will be influenced by the current-induced effects but not by the scalar contributions due to temperature or salinity. TRA performance, in the presence of steady currents, is investigated both theoretically using a simple first order normal mode formulation and numerically using a parabolic equation code for moving media, GCPEM [D. Mikhin, J. Acoust. Soc. Am. 105, 1362 (1999)]. In a multipath shallow ocean environment, the retrofocus field is shifted relative to its location in a nonmoving medium. This shift depends on the current speed and the range-depth dependency of the ocean current profile because each acoustic mode is influenced differently. The possibility of using TRAs for monitoring coastal currents will be discussed.