Due to the rapid spread of COVID-19 and its induced death worldwide, it is imperative to develop a reliable tool for the early detection of this disease. Chest X-ray is currently accepted to be one of the reliable means for such a detection purpose. However, most of the available methods utilize large training data, and there is a need for improvement in the detection accuracy due to the limited boundary segment of the acquired images for symptom identifications. In this study, a robust and efficient method based on transfer learning techniques is proposed to identify normal and COVID-19 patients by employing small training data. Transfer learning builds accurate models in a timesaving way. First, data augmentation was performed to help the network for memorization of image details. Next, five state-of-the-art transfer learning models, AlexNet, MobileNetv2, ShuffleNet, SqueezeNet, and Xception, with three optimizers, Adam, SGDM, and RMSProp, were implemented at various learning rates, 1e-4, 2e-4, 3e-4, and 4e-4, to reduce the probability of overfitting. All the experiments were performed on publicly available datasets with several analytical measurements attained after execution with a 10-fold cross-validation method. The results suggest that MobileNetv2 with Adam optimizer at a learning rate of 3e-4 provides an average accuracy, recall, precision, and F-score of 97%, 96.5%, 97.5%, and 97%, respectively, which are higher than those of all other combinations. The proposed method is competitive with the available literature, demonstrating that it could be used for the early detection of COVID-19 patients.