Ocean current forecast is vital for developing tidal energy and construction of offshore structures in the strait waters. This paper developed a short-term ocean current forecasting approach, which consists of the measured data preprocessing, kernel function selection, and data forecasting using the warped Gaussian process (WGP). A preprocessing using the wavelet thresholding method was proposed to enhance the quality of the measured raw data. The theory of WGP and the commonly used kernel functions were briefly introduced. The sliding time window and one-step ahead strategies were employed to increase the accuracy of predictions. Observations collected during an ocean current measurement campaign executed in a strait water on the coast of the East China Sea were used as an example data set. The current velocity and profile were forecasted and validated using the example data set as an illustration of the framework of the developed approach. The effects of window length, kernel function, and time interval on the WGP forecasting efficiency and precision were investigated. The forecasting performance of the developed WGP model was discussed by comparing it with the standard Gaussian process prediction (GP) model. The current profile with a 95% confidence interval was also predicted by the developed WGP model at a certain point. The validation shows that the developed model is efficient in the short-term ocean current forecast.