We propose a variant of the housing market model à la Shapley and Scarf (in J Math Econ 1:23–37, 1974) that incorporates a limited form of externality in consumption; that is, agents care both about their own consumption (demand preferences) and about the agent who receives their endowment (supply preferences). We consider different domains of preference relations by taking demand and supply aspects of preferences into account. First, for markets with three agents who have (additive) separable preferences such that all houses and agents are acceptable, the strong core is nonempty; a result that can be neither extended to the unacceptable case nor to markets with a larger number of agents. Second, for markets where all agents have demand lexicographic preferences (or all of them have supply lexicographic preferences), we show that the strong core is nonempty, independent of the number of agents and the acceptability of houses or agents, and possibly multi-valued.