Avian A/H5N1 influenza viruses pose a pandemic threat. As few as five amino acid substitutions, or four with reassortment, might be sufficient for mammal-to-mammal transmission by respiratory droplets. From surveillance data we find that two of these substitutions are common in A/H5N1 viruses and thus some viruses might require only three additional substitutions to become transmissible via respiratory droplets between mammals. We use a mathematical model of within-host virus evolution to study factors that could increase and decrease the probability of the remaining substitutions evolving after the virus has infected a mammalian host. These factors combined with the presence of some of these substitutions in circulating strains, make a virus evolving in nature a potentially serious threat. These results highlight critical areas where more data are needed for assessing, and potentially averting, this threat.