The Covid-19 crisis has highlighted the vulnerability of healthcare establishments’ supply chains, highlighting their dependence on a globalised chain that is primarily focused on economic aspects rather than the essential criteria of robustness and sustainability. It is becoming imperative for healthcare establishments to adapt their strategies so as to transcend the current approach focused primarily on efficiency and cost reduction, in favour of an intelligent, integrated supply chain. This must encourage the sharing of information while promoting proactive environmental and social initiatives. This article is the first stage in a research programme designed to provide managers of healthcare institutions with theoretical, methodological and empirical tools to help them in their quest to transition their supply chain towards sustainable development. It also aims to propose reliable and robust measurement methods and tools for determining the economic (minimisation of supply, production, storage and transport costs), environmental (reduction of waste, rationalisation of waste, reduction of GHG emissions, etc.) and social (working conditions and patient satisfaction) benefits arising from the sustainable transition of the supply chain. To do this, we used a narrative literature review. The literature review enabled us to identify nine key elements which, a priori, favour a sustainable supply chain transition in healthcare establishments. We grouped these key elements into major themes, namely the perspective of sustainable development, the prerequisites for a sustainable transition, best practices for a sustainable transition, the performance resulting from the transition, and the learning healthcare systems approach, including the precepts of communicative action. Despite the scarcity of literature on this theme, this study has enabled us to identify key elements, analyse their interactions and highlight existing gaps. These discoveries will guide future research in this field.