With recent advances in computational performance, fluid–structure interaction (FSI) analyses have become feasible within practical computational time. However, in industrial product design, a large number of parametric analyses are required, and therefore further reduction of the computational cost of each individual FSI analysis remains an important challenge. In this study, we focus on the partitioned approach because of its practical advantage of allowing the use of existing solvers. Partitioned approaches are categorized into staggered and iterative schemes. From the viewpoints of accuracy and stability, this study considers the partitioned iterative scheme. Based on these considerations, the primary objective of this research is to develop a method that enables fast and efficient partitioned iterative FSI analyses. Among partitioned approaches, the most widely used method is the Dirichlet–Neumann (D–N) coupling based on the block Gauss–Seidel iteration. However, D–N coupling requires sequential execution of the fluid solver, structural solver, and mesh-update solver, which is disadvantageous in terms of computational efficiency. On the other hand, the block Jacobi method, which allows parallel execution of individual solvers, suffers from slow convergence. To address these issues, this study formulates the FSI problem based on a quasi-Newton framework and proposes a Block-Jacobi-like coupling method. To further accelerate the computation, we introduce a Hyper-Reduced Order Model (HROM). First, using a large set of precomputed snapshots, we construct reduced-order models for the structural analysis, fluid analysis, and mesh-update analysis. By incorporating reduced quadrature, we achieve hyper-reduced order modeling for the entire FSI analysis. The constructed HROM is capable of generating highly accurate FSI approximations at very low computational cost, which serve as efficient initial guesses. In this study, these initial guesses are supplied as a hot start for the full-order model FSI analysis for accelerating the iterative solution process.