The subject of this study is that of coupling the boundary element method (BEM) and a finite element-like interpolation procedure for the analysis of elastic bodies undergoing large deformations. The nonlinear integral relationships for the problem are described and presented in detail according to a total Lagrangian approach. The domain is discretized by quadratic boundary elements and interior cells and the displacements of all interior nodes are calculated from the integral representation. The domain variables, which include the deformation gradients and the 2nd Piola-Kirchhoff stresses, are interpolated through a finite element process. The direct technique whereby the deformation gradients are determined from analytical differentiation of the displacement representation, which requires the integration of higher order singularities, is totally eliminated. This allows for an easier calculation of the domain terms which account for the nonlinear portion of the problem. An iteration procedure is used to solve the integral formulation and numerical calculations are performed for several example problems. In each example, comparison is made with finite element method (FEM) solutions and, whenever possible, with the analytic solutions. These comparisons demonstrate the applicability, effectiveness and limitations of the proposed approach.