1. Subjects requiring ventilatory support (i.e., on mechanical ventilation, non-invasive ventilation, or high-flow nasal cannula) or those subjects who, in the opinion of the investigator, are at risk of imminent respiratory failure (e.g., prolonged respiratory rate >30, signs of respiratory muscle fatigue, or paradoxical diaphragm) 2. Presence or suspicion of active malignancy with the exception of cancer in situ (e.g., skin cancer) 3. Evidence of serious active infections other than COVID-19 4. Current diagnosis of human immunodeficiency virus, hepatitis B or C 5. In the opinion of the investigator, unlikely to survive for > 24 hours from enrollment 6. Women who are pregnant or might be pregnant, or who are currently breast-feeding Subjects must agree to not donate ova or sperm through 30 days after the last dose of study medication. 7. Presence of significant comorbidity that, in the opinion of the investigator, predisposes the subject to mortality. Such conditions might include: a. New York Heart Association class IV Heart Failure b. Hepatic dysfunction (i.e., AST or ALT >3x upper limit of normal) c. Renal dysfunction (i.e., estimated glomerular filtration rate (eGFR) <50 mL/min) or receiving renal replacement therapy 8. Presence of septic shock at time of enrollment 9. Hemoglobin < 80 g/L 10. Evidence of neutropenia (i.e., absolute neutrophil count < 1000 cells/μL), lymphopenia (i.e., absolute lymphocyte count < 500 cells/μL) or thrombocytopenia (i.e., platelets < 50×109/μL) 11. Hypersensitivity to TD-0903 or its components, or to other JAK inhibitors 12. Treatment with anti-IL 6, anti-IL-6R antagonists, or with JAK inhibitors in the past 30 days, or plans to receive a JAK inhibitor during the study period 13. Current treatment with conventional synthetic disease-modifying anti-rheumatic drugs (DMARDs)/immunosuppressive agents including: a. Methotrexate, cyclosporine, mycophenolate, tacrolimus, penicillamine, or sulfasalazine within 2 weeks prior to enrollment b. Azathioprine or cyclophosphamide within 12 weeks prior to enrollment c. Tumor Necrosis Factor-alpha (TNFα) inhibitors within 12 weeks prior to enrollment 14. Participating in other clinical trials involving any other experimental treatment for COVID-19, except in the context of a single-arm antiviral compassionate-use protocol 15. Use of chronic oral corticosteroids for a non-COVID-19-related condition in a dose higher than prednisone 5 mg or equivalent per day 16. Subjects with active or incompletely treated pulmonary tuberculosis, or known history of non-tuberculosis mycobacterium over past 12 months 17. Subject requires continuous oxygen supplementation for underlying cardiorespiratory history in the past 90 days 18. Body Mass Index ≥40 kg/cm2 19. Receipt of any live vaccine in the 4 weeks prior to visit 1 or plans to receive a live vaccine during the study period 20. History of venous thromboembolism (VTE), deep venous thrombosis (DVT), Pulmonary Embolism (PE) or hypercoagulable state.

1. Subjects requiring ventilatory support (i.e., on mechanical ventilation, non-invasive ventilation, or high-flow nasal cannula) or those subjects who, in the opinion of the investigator, are at risk of imminent respiratory failure (e.g., prolonged respiratory rate >30, signs of respiratory muscle fatigue, or paradoxical diaphragm) 2. Presence or suspicion of active malignancy with the exception of cancer in situ (e.g., skin cancer) 3. Evidence of serious active infections other than COVID-19 4. Current diagnosis of human immunodeficiency virus, hepatitis B or C 5. In the opinion of the investigator, unlikely to survive for > 24 hours from enrollment 6. Women who are pregnant or might be pregnant, or who are currently breast-feeding Subjects must agree to not donate ova or sperm through 30 days after the last dose of study medication. 7. Presence of significant comorbidity that, in the opinion of the investigator, predisposes the subject to mortality. Such conditions might include: a. New York Heart Association class IV Heart Failure b. Hepatic dysfunction (i.e., AST or ALT >3x upper limit of normal) c. Renal dysfunction (i.e., estimated glomerular filtration rate (eGFR) <50 mL/min) or receiving renal replacement therapy 8. Presence of septic shock at time of enrollment 9. Hemoglobin < 80 g/L 10. Evidence of neutropenia (i.e., absolute neutrophil count < 1000 cells/μL), lymphopenia (i.e., absolute lymphocyte count < 500 cells/μL) or thrombocytopenia (i.e., platelets < 50×109/μL) 11. Hypersensitivity to TD-0903 or its components, or to other JAK inhibitors 12. Treatment with anti-IL 6, anti-IL-6R antagonists, or with JAK inhibitors in the past 30 days, or plans to receive a JAK inhibitor during the study period 13. Current treatment with conventional synthetic disease-modifying anti-rheumatic drugs (DMARDs)/immunosuppressive agents including: a. Methotrexate, cyclosporine, mycophenolate, tacrolimus, penicillamine, or sulfasalazine within 2 weeks prior to enrollment b. Azathioprine or cyclophosphamide within 12 weeks prior to enrollment c. Tumor Necrosis Factor-alpha (TNFα) inhibitors within 12 weeks prior to enrollment 14. Participating in other clinical trials involving any other experimental treatment for COVID-19, except in the context of a single-arm antiviral compassionate-use protocol 15. Use of chronic oral corticosteroids for a non-COVID-19-related condition in a dose higher than prednisone 5 mg or equivalent per day 16. Subjects with active or incompletely treated pulmonary tuberculosis, or known history of non-tuberculosis mycobacterium over past 12 months 17. Subject requires continuous oxygen supplementation for underlying cardiorespiratory history in the past 90 days 18. Body Mass Index ≥40 kg/cm2 19. Receipt of any live vaccine in the 4 weeks prior to visit 1 or plans to receive a live vaccine during the study period 20. History of venous thromboembolism (VTE), deep venous thrombosis (DVT), Pulmonary Embolism (PE) or hypercoagulable state.