Abstract Polymeric nanostructured biomaterials can be used as synthetic cell interfaces with important applications in the study and control of cellular processes. Herein, we developed multifunctional nanocomposites based on synthesized biodegradable and biocompatible copolyesters of poly(butylene 1,4-trans-cyclohexanedicarboxylate) (PBCE) containing ether–linkages, and single walled carbon nanotubes (SWCNTs), employed as functional phase. Surface, thermal and mechanical characterization of the polymer and nanocomposite films were performed. The influences of AC conductivity and interfacial polarization on dielectric relaxation process, as well as the correlation between the dielectric behaviors and SWCNT content were investigated by measuring the dielectric properties. The effect of SWCNT incorporation, and amount of ether-oxygen atoms was also investigated in terms of fibroblast long-term culture stability, by performing adhesion and proliferation studies of cells seeded on the biomaterial surface, at different time points. Results showed that polymeric conductive nanocomposites were successfully developed with a low percolation threshold, and SWCNT presence maintained the polymer thermal degradation behavior. Moreover, the culture of primary fibroblasts indicated that these advanced functional materials are biocompatible and guarantee the cell adhesion and growth, being suitable substrates for regenerative medicine applications. Finally, their versatile structure and chemical properties may provide a robust platform to gain insight into cell–biomaterial interactions, being an important step towards the better understanding and control of cell interactions with nanomaterials.