Background: In the current COVID-19 pandemic, with an absence of approved drugs and widely accessible vaccines, repurposing existing drugs is vital to quickly developing a treatment for the disease. Methods: In this study, we used a dataset consisting of sequences of viral proteins and chemical structures of pharmaceutical drugs for known drug–target interactions (DTIs) and artificially generated non-interacting DTIs to train a binary classifier with the ability to predict new DTIs. Random Forest (RF), deep neural network (DNN), and convolutional neural networks (CNN) were tested. The CNN and RF models were selected for the classification task. Results: The models generalized well to the given DTI data and were used to predict DTIs involving SARS-CoV-2 nonstructural proteins (NSPs). We elucidated (with the CNN) 29 drugs involved in 82 DTIs with a 97% probability of interaction, 44 DTIs of which had a 99% probability of interaction, to treat COVID-19. The RF elucidated 6 drugs involved in 17 DTIs with a 90% probability of interacting. Conclusions: These results give new insight into possible inhibitors of the viral proteins beyond pharmacophore models and molecular docking procedures used in recent studies.