The mechanical response of a contractile cell anchored to the substrate through focal adhesions is studied by means of an asymmetric pre-strained tensegrity structure obeying a neo-Hookean stress–strain law. The aim is to assess the influence of overall asymmetric contraction on the cell durotaxis and on the growth of the focal adhesion plaque. The asymmetric kinematics of the system is obtained in two ways, that is by assuming a gradient of the substrate stiffness and through asymmetric buckling. Equivalent springs are purposely considered to represent the stiffness of the ensemble formed by the substrate, the focal adhesion plaque and the integrin ligands. Then, contraction results from elastic strains induced by competing polymerization and actomyosin contraction. The cell mechanical response in terms of durotaxis and its coupling with focal adhesion plaque growth is finally analysed with respect to the effects of asymmetry, gaining some insights into how this asymmetry could participate to redirect cell migration, both in terms of durotaxis and mollitaxis.