Large outbreaks of zoonotic influenza A virus (IAV) infections may presage an influenza pandemic. However, the likelihood that an airborne-transmissible variant evolves upon zoonotic infection or co-infection with zoonotic and seasonal IAVs remains poorly understood, as does the relative importance of accumulating mutations versus re-assortment in this process. Using discrete-time probabilistic models, we determined quantitative probability ranges that transmissible variants with 1–5 mutations and transmissible re-assortants evolve after a given number of zoonotic IAV infections. The systematic exploration of a large population of model parameter values was designed to account for uncertainty and variability in influenza virus infection, epidemiological and evolutionary processes. The models suggested that immunocompromised individuals are at high risk of generating IAV variants with pandemic potential by accumulation of mutations. Yet, both immunocompetent and immunocompromised individuals could generate high viral loads of single and double mutants, which may facilitate their onward transmission and the subsequent accumulation of additional 1–2 mutations in newly-infected individuals. This may result in the evolution of a full transmissible genotype along short chains of contact transmission. Although co-infection with zoonotic and seasonal IAVs was shown to be a rare event, it consistently resulted in high viral loads of re-assortants, which may facilitate their onward transmission among humans. The prevention or limitation of zoonotic IAV infection in immunocompromised and contact individuals, including health care workers, as well as vaccination against seasonal IAVs—limiting the risk of co-infection—should be considered fundamental tools to thwart the evolution of a novel pandemic IAV by accumulation of mutations and re-assortment.