Even though there has been extensive research in computational methods for solving Fluid-Structure interaction (FSI) problems in the last decades, a comprehensive presentation from a mathematical point of view is still missing nowadays: one of the reasons for this is the fact that the two subproblems, namely the Navier–Stokes equation and the elastic solid equation, are two big mathematical challenges on their own. FSI problems describe the dynamic interplay between a fluid and a solid. This interplay takes place because of the coupling of the two different physics at the FSI interface, namely the part of the physical domain that is common to the fluid subdomain and the solid subdomain. The FSI interface profile is unknown a priori and depends on the dynamics of the fluid and the structure problem. Our work aims to introduce a framework where Domain Decomposition (DD) algorithms and Reduced-Order Models (ROMs) are combined in order to achieve better performance of numerical simulations. We choose to model the DD using an optimisation approach to ensure the coupling of the interface conditions between fluid and structure subdomains and it leads to a complete separation of the solvers on the subdomains. The snapshots for the high–fidelity model are obtained with the Finite Element (FE) discretisation, and the model order reduction is then proposed both in terms of time and physical parameters with a standard Proper Orthogonal Decomposition (POD)-Galerkin projection. Despite the fact that the partitioned approach is more attractive because of its computational efficiency, it might lead to unstable algorithms, under some physical and geometrical conditions, while the monolithic approach does not suffer from this issue. This happens, for example, if the physical domain has a slender shape, or, as in our numerical test, if the fluid density is close to the solid density, and this is usually the case in haemodynamics applications, where the density of the blood is quite close to the density of the walls of the vessel. This phenomenon is the so-called "added mass" effect. The approach undertaken in our work leads to a fully segregated algorithm which is nevertheless stable under the "added mass'" effect. It is evidenced by the numerical experiments of the model presented for a two–dimensional haemodynamics benchmark FSI problem.