Developing new barrier membranes with improved biomechanical characteristics has acquired much interest owing to their crucial role in the field of periodontal tissue regeneration. In this regard, we enriched the electrospun polycaprolactone (PCL)/gelatin (Gel) membranes by adding bioglass (BG) or Cu-doped bioglass (CuBG) and examined their cellular adhesion and proliferation potential in the presence of alveolar bone marrow-derived mesenchymal stem cells (aBMSCs). The membranes were fabricated and characterized using mechanical strength, SEM, FTIR, EDX, and ICP assay. Besides, aBMSCs were isolated, characterized, and seeded with a density of 35,000 cells in each experimental group. Next, the cellular morphology, cell adhesion capacity, proliferation rate, and membrane antibacterial activity were assessed. The results displayed a significant improvement in the wettability, pore size, and Young’s modulus of the PCL membrane following the incorporation of gelatin and CuBG particles. Moreover, all scaffolds exhibited reasonable biocompatibility and bioactivity in physiological conditions. Although the PCL/Gel/CuBG membrane revealed the lowest primary cell attachment, cells were grown properly and reached the confluent state after seven days. In conclusion, we found a reasonable level of attachment and proliferation of aBMSCs on all modified membranes. Meanwhile, a trace amount of Cu provided superiority for PCL/Gel/CuBG in periodontal tissue regeneration.