This paper discusses the notion of ``core bio-inspired services" - low-level services providing basic bio-inspired mechanisms, such as evaporation, aggregation or spreading - shared by higher-level services or applications. Design patterns descriptions of self-organising mechanisms, such  as gossip, morphogenesis, or foraging, show that these higher-level mechanisms are composed of basic bio-inspired mechanisms (e.g. digital pheromone is composed of spreading, aggregation and evaporation). In order to ease design and implementation of self-organising applications (or high-level services), by supporting reuse of code and algorithms, this paper proposes to provide these basic services at the heart of any middleware or infrastructure supporting such applications, and to provide them as "core" built-in services around which all other services are built. The idea of those "core services" is similar to natural behaviour, where the environment itself is responsible to provide some services (e.g. evaporation). Such a core, called here BIO-CORE, is typically part of a  middleware or a supporting infrastructure, and provides those distributed services (as an environment would do) for all other higher-level services and applications, which then share and (re-)use them. This paper describes the core design as well as preliminary simulations showing the feasibility of the concept.