Abstract There is an increasing interest in researching urban invasion. When invasive species establish and spread in urban areas, their management becomes particularly challenging given the influence of public presence and perceptions. Cultural, religious, and moral values play a crucial role in shaping public perceptions toward management strategies for invasive species. Free‐ranging dogs (FRDs) and free‐ranging cats (FRCs) are among the urban invasive species that bring health, safety, and wildlife conservation concerns. However, given their close relationship with people controlling their populations is challenging. We surveyed 500 residents of Karaj city in northern Iran to understand the behaviour of feeding FRDs and FRCs by people, as well as to uncover residents' acceptability of different control measures, focusing on culling FRDs and FRCs under different scenarios. We used generalized linear mixed models and cumulative link mixed models to address the issues of feeding FRDs and FRCs and acceptability of culling them respectively. We found that more than half of the respondents fed FRDs and FRCs (63% and 59%, respectively), and among those who fed FRDs and FRCs, 74.8% and 78.2% respectively used food leftovers from home to feed these animals. Furthermore, around half and one‐third of respondents provided FRDs and FRCs outside their neighbourhoods, respectively. We also found that the religious belief of dogs being ritually impure (Najis) was a significant predictor for both feeding and culling FRDs. People who held the belief that dogs are Najis were more inclined toward culling and less willing to feed FRDs. Moreover, we found polarized and conflicting views concerning culling in higher‐intensity scenarios involving attacks on people and disease spread. On the contrary, in less‐intense scenarios such as attacks on wildlife and population control in urban areas, most respondents disagreed with culling. In the end, we recommend (1) an improved waste management system for food residuals and leftovers that provide resources for FRDs and FRCs, (2) effective communication with pet‐owners to reduce feeding FRDs and FRCs and (3) bottom‐up de‐escalation of the conflict between polarized groups about culling FRDs and FRCs before it reaches to a destructive phase. Read the free Plain Language Summary for this article on the Journal blog.