Polyhydroxyalkanoates (PHAs) are biodegradable polyesters produced by microorganisms, under unbalanced growth conditions, as a carbon storage compound. PHAs are composed of various monomers such as 3-hydroxybutyrate (3HB) and 3-hydroxyhexanoate (3HHx). Silk fibroin (SF) derived from Bombyx mori cocoons, is a widely studied protein polymer commonly used for biomaterial applications. In this study, non-woven electrospun films comprising a copolymer of 3HB and 3HHx [P(3HB-co-3HHx)], SF and their blends were prepared by electrospinning technique. The growth and osteogenic differentiation of human umbilical cord-derived mesenchymal stem cells (hUC-MSCs) were studied using different types of fabricated electrospun films. The differentiation study revealed that electrospun P(3HB-co-3HHx)/SF film supports the differentiation of hUC-MSCs into the osteogenic lineage, confirmed by histological analysis using Alizarin Red staining, energy dispersive X-ray (EDX) and quantitative real-time PCR analysis (qPCR). Electrospun P(3HB-co-3HHx)/SF film up-regulated the expression of osteogenic marker genes, alkaline phosphatase (ALP) and osteocalcin (OCN), by 1.6-fold and 2.8-fold respectively, after 21 days of osteogenic induction. In conclusion, proliferation and osteogenic differentiation of hUC-MSCs were enhanced through the blending of P(3HB-co-3HHx) and SF. The results from this study suggest that electrospun P(3HB-co-3HHx)/SF film is a promising biomaterial for bone tissue engineering.