Nanofibers as niche-biomimetic scaffolds exhibit potential in bone tissue engineering (BTE). Here, poly(3-hydroxybutyrate-co-4-hydroxybutyrate) co-polymer (P34HB)/poly(ethylene glycol) (PEG) nanofiber membranes with a high hydrophilicity and mechanical properties were fabricated by introducing PEG to P34HB via electrospinning. The P34HB/PEG nanofibrous scaffolds were investigated for their potential in the osteogenic differentiation and mineralization of bone marrow mesenchymal stem cells (BMSCs). By adjusting the ratio of PEG to P34HB, three scaffolds, including P34HB, P34HB/10 wt%PEG, and P34HB/30 wt%PEG, were successfully fabricated. The composite P34HB/PEG nanofiber membranes showed an enhanced hydrophilicity, a decreased fiber size, and an increased mechanical strength compared with those of P34HB. In-vitro studies showed that the P34HB/PEG membranes better supported cell adhesion, spreading, and proliferation than those of P34HB. The incorporation of PEG into the P34HB scaffold also promoted the osteoinduction capacity, as evidenced by activation of the alkaline phosphatase activity (ALP) activity, increased gene expression of bone specific markers (such as RUNX2, ALP, Col1a1, OPN, OCN, and BMP2), and mineral nodules formation. Comparatively, P34HB/10 wt%PEG showed a higher hydrophilicity and mechanical properties, as well as a better biological performance than the other membranes. Thus, the electrospun P34HB/PEG nanofiber membranes may be potentially developed as regenerative materials for BTE applications.