The prevention of chronic wound formation has already been a primary subject in wound management, particularly for deep wounds. The electrospun nanofiber membranes hold tremendous potential in the prevention of chronic wounds due to their micro/nano pore structures. Currently, many natural and synthetic materials have been utilized in the fabrication of nanofiber membranes. However, striking a balance between the structural stability and the biocompatibility remains challenging. It is necessary not only to ensure the long-term durability of nanofiber membranes but also to enhance their biocompatibility for alleviating patients' suffering. In this study, we reported a nanofiber membrane dressing with excellent biocompatibility and mechanical properties, which is potential for the treatment of deep wounds. The basal material chosen for the preparation of the nanofiber membrane was a co-polyester (NI-LPGD5) synthesized by non-isocyanate polyurethane (NIPU) and polyglycolic acid with a dihydroxy structure (LPGD—synthesized from glycolic acid and neopentyl glycol). Moreover, curcumin was also added as a bioactive substance to enhance the pro-healing effect of dressings. The physicochemical properties of the prepared nanofiber membranes were characterized through various physicochemical tools. Our results demonstrated that the NI-LPGD5 co-polymer can be electrospun into smooth fibers. Meanwhile, curcumin-loaded nanofiber membranes (Cur/NI-LPGD5) also exhibited a favorable microscopic morphology. The fabricated membranes exhibited suitable mechanical properties, outstanding hygroscopic-swelling rate and water vapor transmittance. Besides, in vitro cell culturing, the cells on the NI-LPGD5 membrane maintained their maximum viability. The potential of in vivo wound healing was further demonstrated through animal experiments. The experimental results showed that the nanofiber membranes effectively prevented chronic wounds from forming and promoted granulation tissue growth without replacing the dressing throughout the healing process. We also found that these nanofiber membranes could effectively promote the expression of related biomarkers to accelerate wound healing, particularly the Cur/NI-LPGD5 membrane. In conclusion, the fabricated membranes possess suitable physicochemical properties and promising bioactivity. As a result, it effectively prevented the formation of chronic wounds and demonstrated significant potential in reducing the frequency of dressing changes.