Two elementary models of ocean circulation, the well-known double-gyre stream function model and a single-layer quasi-geostrophic (QG) basin model, are used to generate flow data that sample a range of possible dynamical behavior for particular flow parameters. A reservoir computing (RC) machine learning algorithm then learns these models from the stream function time series. In the case of the QG model, a system of partial differential equations with three physically relevant dimensionless parameters is solved, including Munk- and Stommel-type solutions. The effectiveness of a RC approach to learning these ocean circulation models is evident from its ability to capture the characteristics of these ocean circulation models with limited data including predictive forecasts. Further assessment of the accuracy and usefulness of the RC approach is conducted by evaluating the role of both physical and numerical parameters and by comparison with particle trajectories and with well-established quantitative assessments, including finite-time Lyapunov exponents and proper orthogonal decomposition. The results show the capability of the methods outlined in this article to be applied to key research problems on ocean transport, such as predictive modeling or control.