Bimetallic nanoparticles (often known as nanoalloys) with core-shell arrangement are of special interest in several applications, such as in optics, catalysis, magnetism and biomedicine. Despite wide interest in applications, the physical factors stabilizing the structures of these nanoparticles are still unclear to a great extent, especially for what concerns the relationship between geometric structure and chemical ordering pattern. Here global-optimization searches are performed in order to single out the most stable chemical ordering patterns corresponding to the most important geometric structures, for a series of weakly miscible systems, including AgCu, AgNi, AgCo and AuCo. The calculations show that (i) the overall geometric structure of the nanoalloy and the shape and placement of its inner core are strictly correlated; (ii) centered cores can be obtained in icosahedral nanoparticles but not in crystalline or decahedral ones, in which asymmetric quasi-Janus morphologies form; (iii) in icosahedral nanoparticles, when the core exceeds a critical size, a new type of morphological instability develops, making the core asymmetric and extending it towards the nanoparticle surface; (iv) multi-center patterns can be obtained in polyicosahedral nanoalloys.Analogies and differences between the instability of the core in icosahedral nanoalloys and the StranskiKrastanov instability occurring in thin-film growth are discussed. All these issues are crucial for designing strategies to achieve effective coatings of the cores.a Corresponding author,