Abstract In this paper we analyze the numerical oscillations affecting loosely coupled schemes for hybrid-dimensional 0D–3D fluid–structure interaction (FSI) problems, which arise e.g. in the field of cardiovascular modeling, and we propose a novel stabilized scheme that cures this issue. We study several loosely coupled schemes, including the Dirichlet–Neumann (DN) and Neumann–Dirichlet (ND) schemes. In the first one, the 0D fluid model prescribes the pressure to the 3D structural mechanics model and receives the flow. In the second one, on the contrary, the fluid model receives the pressure and prescribes the flow. The terms DN and ND, employed in the FSI literature, are borrowed from domain decomposition methods, although here a single iteration is performed before moving on to the next time step (that is, the coupling is treated explicitly). Should the fluid be enclosed in a cavity, the DN scheme is affected by non-physical oscillations whose origin lies in the balloon dilemma, for which we provide an algebraic interpretation. Moreover, we show that also the ND scheme can be unstable for a range of parameter choices. Surprisingly, increasing either the viscous dissipation or the inertia of the structure favours the onset of oscillations and, for certain parameter choices, the ND is unconditionally unstable. In the presence of inertial terms, by reducing the time step size below a certain threshold, the amplitude of the numerical oscillations is even amplified. We provide an explanation for these facts and establish sharp stability bounds on the time step size. Our analysis extends to Robin–Robin schemes, based on linear combinations of the conditions of pressure continuity and either volume or flux continuity. While appropriate choices of Robin coefficients can achieve numerical stability, tuning these coefficients can be challenging in practice. To address these issues, we propose a numerically consistent stabilization term for the Neumann–Dirichlet scheme, inspired by physical insight on the onset of oscillations. We prove that our proposed stabilized scheme is absolutely stable for any choice of time step size. Notably, the proposed scheme does not require parameter tuning. These results are verified by several numerical tests. Finally, we apply the proposed stabilized scheme to an important problem in cardiac electromechanics, namely the coupling between a 3D cardiac model and a closed-loop lumped-parameter model of blood circulation. In this setting, our proposed scheme successfully removes the non-physical oscillations that would otherwise affect the numerical solution.