Social insects are very successful invasive species, and the continued increase of global trade and transportation has exacerbated this problem. The yellow-legged hornet, Vespa velutina nigrithorax (henceforth Asian hornet), is drastically expanding its range in Western Europe. As an apex insect predator, this hornet poses a serious threat to the honey bee industry and endemic pollinators. Current suppression methods have proven too inefficient and expensive to limit its spread. Gene drives might be an effective tool to control this species, but their use has not yet been thoroughly investigated in social insects. Here, we built a model that matches the hornet's life history and modelled the effect of different gene drive scenarios on an established invasive population. To test the broader applicability and sensitivity of the model, we also incorporated the invasive European paper wasp Polistes dominula. We find that although a gene drive can spread through a social wasp population, it can only do so under stringent gene drive-specific conditions. The main issue is that the large number of offspring that social wasp colonies produce guarantees that, even with very limited formation of resistance alleles, such alleles will quickly spread and rescue the population. Furthermore, we find that only a gene drive targeting female fertility is promising for population control due to the haplodiploidy of social insects. Nevertheless, continued improvements in gene drive technology may make it a promising method for the control of invasive social insects.