Deep neural operator modeling frameworks have been recently developed for forecasting time series data. Such machine learning models may be useful in multiple domains including the atmospheric and oceanic ones, and in general, the larger fluids community. The present work investigates the effectiveness of deep neural operator models for reproducing and predicting fluid flows and realistic ocean simulations. We first evaluate the capabilities of such deep neural operator models when trained on simulated two-dimensional fluid flows. We then investigate their application to forecasting ocean circulation in the Gulf of Mexico, Middle Atlantic Bight, and Massachusetts Bay, learning from high-resolution data-assimilative simulations employed for real sea experiments. We confirm that trained deep neural operator models are capable of predicting idealized periodic eddy shedding. For realistic ocean surface flows, they can predict several of the features and show some skill, providing potential for applications.