The purpose of this study was to evaluate the growth patterns and osteogenic differentiation of human bone marrow mesenchymal stem cells (hBMSCs) when seeded onto new biodegradable chitosan/polyester scaffolds. Scaffolds were obtained by melt blending chitosan with poly(butylene succinate) in a proportion of 50% (wt) each and further used to produce a fiber mesh scaffold. hBMSCs were seeded on those structures and cultured for 3 weeks under osteogenic conditions. Cells were able to reduce MTS and demonstrated increasing metabolic rates over time. SEM observations showed cell colonization at the surface as well as within the scaffolds. The presence of mineralized extracellular matrix (ECM) was successfully demonstrated by peaks corresponding to calcium and phosphorus elements detected in the EDS analysis. A further confirmation was obtained when carbonate and phosphate group peaks were identified in Fourier Transformed Infrared (FTIR) spectra. Moreover, by reverse transcriptase (RT)-PCR analysis, it was observed the expression of osteogenic gene markers, namely, Runt related transcription factor 2 (Runx2), type 1 collagen, bone sialoprotein (BSP), and osteocalcin. Chitosan-PBS (Ch-PBS) biodegradable scaffolds support the proliferation and osteogenic differentiation of hBMSCs cultured at their surface in vitro, enabling future in vivo testing for the development of bone tissue engineering therapies.