Emerging diseases are a major challenge to public health. Revealing the evolutionary processes that allow novel pathogens to adapt to new hosts, also the potential barriers to host adaptation, is central to understanding the drivers of disease emergence. In particular, it is unclear how the genetics and ecology of pathogens interact to shape the likelihood of successful cross-species transmission. To better understand the determinants of host adaptation and emergence, we modelled key aspects of pathogen evolutionary dynamics at both intra- and inter-host scales, using parameter values similar to those observed in influenza virus. We considered the possibility of acquiring the necessary host adaptive mutations both before (‘off-the-shelf’ emergence) and after (‘tailor-made’ emergence) a virus is transmitted from a donor to a new recipient species. Under both scenarios, population bottlenecks at inter-host transmission act as a major barrier to host adaptation, greatly limiting the number of adaptive mutations that are able to cross the species barrier. In addition, virus emergence is hindered if the fitness valley between the donor and recipient hosts is either too steep or too shallow. Overall, our results reveal where in evolutionary parameter space a virus could adapt to and become transmissible in a new species.