The fluid–structure interaction (FSI) problems with large deformations remain challenging in terms of numerical accuracy and stability. Since mesh motion and remeshing are not required, fully Eulerian approaches have attracted attention as an alternative to Lagrangian and Euler–Lagrangian methods. In fully Eulerian FSI methods, both the fluid and structural phases are described in a fixed coordinate system. The governing equations consist of the Navier–Stokes equations and transport equations for the volume fraction and deformation tensor. While this formulation enables the natural representation of large structural deformations, a critical difficulty arises from numerical diffusion of the deformation tensor into the fluid region. This leakage may cause the deformation tensor to grow exponentially, leading to unphysical stresses and numerical instability. As the deformation tensor is smooth in the structural phase and discontinuous across the fluid–structure interface, a numerical method capable of handling both smooth and discontinuous solutions is required. In this study, we incorporate the Boundary Variation Diminishing (BVD) method into the fully Eulerian FSI framework to improve numerical accuracy and stability. The BVD method was originally developed for compressible flows containing smooth and discontinuous solutions, and dynamically selects among different spatial reconstruction methods to capture such complex distributions. Taking advantage of this, we suppress the unphysical diffusion of the deformation tensor at the fluid–structure interface. We demonstrate the performance of the proposed method solving a cavity flow involving an elastically deforming structure. Conventional methods suffer from loss of stability due to excessive numerical dissipation, whereas the BVD method allows the computation to proceed stably. Moreover, since the fully Eulerian approach is well suited for voxel-based geometries obtained from medical imaging, the proposed method is expected to be applicable to biomechanical FSI problems such as long-term transport of deformable red blood cells.