As acute infectious pneumonia, the coronavirus disease-2019 (COVID-19) has created unique challenges for each nation and region. Both India and the United States (US) have experienced a second outbreak, resulting in a severe disease burden. The study aimed to develop optimal models to predict the daily new cases, in order to help to develop public health strategies. The autoregressive integrated moving average (ARIMA) models, generalised regression neural network (GRNN) models, ARIMA–GRNN hybrid model and exponential smoothing (ES) model were used to fit the daily new cases. The performances were evaluated by minimum mean absolute per cent error (MAPE). The predictive value with ARIMA (3, 1, 3) (1, 1, 1)14 model was closest to the actual value in India, while the ARIMA–GRNN presented a better performance in the US. According to the models, the number of daily new COVID-19 cases in India continued to decrease after 27 May 2021. In conclusion, the ARIMA model presented to be the best-fit model in forecasting daily COVID-19 new cases in India, and the ARIMA–GRNN hybrid model had the best prediction performance in the US. The appropriate model should be selected for different regions in predicting daily new cases. The results can shed light on understanding the trends of the outbreak and giving ideas of the epidemiological stage of these regions.