Subiteration forms the basic iterative method for solving the aggregated equations in fluid-structure-interaction problems, in which the fluid and structure equations are solved alternately subject to complementary partitions of the interface conditions. However, this subiteration process can be defective or inadequate, as it is endowed with only conditional stability and, moreover, divergence can occur despite formal stability due to nonnormality. Furthermore, the subiteration method generally operates within a sequential time-integration process to solve a sequence of similar problems, but is unable to exploit this property. To overcome these shortcomings, the present work proposes to accelerate the subiteration method by means of a Krylov method, viz., GMRES. We show that the Krylov space can be composed of vectors in a low-dimensional subspace associated with the discrete representation of a function on the fluid-structure interface. The corresponding Interface-GMRES-acceleration procedure requires negligible computational resources, and retains the modularity of the underlying subiteration method. Moreover, the Krylov space can be optionally reused in subsequent invocations of the GMRES method, conforming to the GMRESR procedure. Detailed numerical results for a prototypical model problem are presented to illustrate the effectiveness of the proposed Interface GMRES(R)-acceleration of the subiteration method.