This paper presents an initial study of the computational modeling for determining the stress–strain behavior of the cord–rubber segment, tailored to be used as a flexible member in the construction of torque transfer coupling in railway vehicles. The presented computational models use the multibody simulation approach for the assessment of segment deformation under diverse loading characteristics. The results are then used in the finite element computational models of the cord–rubber segment. The computational model is validated through tension tests of cord–rubber segments. The homogenized and microstructural models show their ability to simulate the overall stiffness of the segment relatively accurately in tension, but they are limited in not being able to include the influence of the segment’s manufacturing technology. The micro-scale computational model then incorporates realistic representations of cord bundles, considering various cross-sectional shapes, whereas a very important factor significantly influencing the correlation of the results of the technical experiment and simulation is the consideration of the manufacturing technology in the process of building the computational model. The results show that the computational modeling approach in this work can be used for the determination of the optimal manufacturing technology conditions with regard to the stress–strain behavior of the given segment.