The early and reliable detection of COVID-19 infected patients is essential to prevent and limit its outbreak. The PCR tests for COVID-19 detection are not available in many countries, and also, there are genuine concerns about their reliability and performance. Motivated by these shortcomings, this article proposes a deep uncertainty-aware transfer learning framework for COVID-19 detection using medical images. Four popular convolutional neural networks (CNNs), including VGG16, ResNet50, DenseNet121, and InceptionResNetV2, are first applied to extract deep features from chest X-ray and computed tomography (CT) images. Extracted features are then processed by different machine learning and statistical modeling techniques to identify COVID-19 cases. We also calculate and report the epistemic uncertainty of classification results to identify regions where the trained models are not confident about their decisions (out of distribution problem). Comprehensive simulation results for X-ray and CT image data sets indicate that linear support vector machine and neural network models achieve the best results as measured by accuracy, sensitivity, specificity, and area under the receiver operating characteristic (ROC) curve (AUC). Also, it is found that predictive uncertainty estimates are much higher for CT images compared to X-ray images.