Respiratory viruses continue to afflict mankind. Among them, pathogens such as coronaviruses [including the current pandemic agent known as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)] and the one causing influenza A (IAV) are highly contagious and deadly. These can evade the immune system defenses while causing a hyper inflammatory response that can damage different tissues/organs. Simultaneously targeting several immunomodulatory proteins is a plausible antiviral strategy since it could lead to the discovery of indirect-acting pan-antiviral (I-APA) agents for the treatment of diseases caused by respiratory viruses. In this context, computational approaches, which are an essential part of the modern drug discovery campaigns, could accelerate the identification of multi-target immune modulators. This perspective discusses the usefulness of computational multi-target drug discovery for the virtual screening (drug repurposing) of IAPA agents capable of boosting the immune system through the activation of the toll-like receptor 7 (TL-R7) and/or the stimulator of interferon genes (STING) while inhibiting key inflammation-related proteins such as caspase-1 and tumor necrosis factor-alpha (TNF-α).