The recent COVID-19 outbreak has severely affected people around the world. There is a need of an efficient decision making tool to improve awareness about the spread of COVID-19 infections among the common public. An accurate and reliable neural network based tool for predicting confirmed, recovered and death cases of COVID-19 can be very helpful to the health consultants for taking appropriate actions to control the outbreak. This paper proposes a novel Nonlinear Autoregressive (NAR) Neural Network Time Series (NAR-NNTS) model for forecasting COVID-19 cases. This NAR-NNTS model is trained with Scaled Conjugate Gradient (SCG), Levenberg Marquardt (LM) and Bayesian Regularization (BR) training algorithms. The performance of the proposed model has been compared by using Root Mean Square Error (RMSE), Mean Square Error (MSE) and correlation co-efficient i.e. R-value. The results show that NAR-NNTS model trained with LM training algorithm performs better than other models for COVID-19 epidemiological data prediction.