Since the new coronavirus (COVID-19) outbreak spread from China to other countries, it has been a curiosity for how and how long the number of cases will increase. This study aims to forecast the number of confirmed cases of COVID-19 in Italy, the United Kingdom (UK) and the United States of America (USA). In this study, grey model (GM(1,1)), nonlinear grey Bernoulli model (NGBM(1,1)) and fractional nonlinear grey Bernoulli model (FANGBM(1,1)) are compared for the prediction. Therefore, grey prediction models, especially the fractional accumulated grey model, are used for the first time in this topic and it is believed that this study fills the gap in the literature. This model is applied to predict the data for the period 19/03-22/04/2020 (35 days) and forecast the data for the period 23/04-22/05/2020. The number of cases of COVID-19 in these countries are handled cumulatively. The prediction performance of the models is measured by the calculation of root mean square error (RMSE), mean absolute percentage error (MAPE) and R² values. It is obtained that FANGBM(1,1) gives the highest prediction performance with having the lowest RMSE and MAPE values and the highest R² values for these countries. Results show that the cumulative number of cases for Italy, UK and USA is forecasted to be about 233000, 184000 and 1090000, respectively, on May 22, 2020 which corresponds to the average daily rate is 0.80%, 1.19% and 1.13%, respectively, from 22/04/2020 to 22/05/2020. The FANGBM(1,1) presents that the cumulative number of cases of COVID-19 increases at a diminishing rate from 23/04/2020 to 22/05/2020 for these countries.