Mineralized, bio-inspired nanofibrous scaffolds with controllable architecture and capable of mimicking essential characteristics of bone extracellular matrix at the micro- and nanoscale offer a promising strategy to restore functions or elicit favorable responses for bone tissue regeneration and repair. In this study, a simple approach to fabricate a hybrid scaffold with porous fibers for bone tissue engineering is presented. Non-woven multifunctional nano- and microfibers were fabricated using a block copolymer of poly(ε-caprolactone) (PCL) and poly(lactic acid) (PLA) (PL-b-CL) as a matrix and hydroxyapatite (HA) as a functional agent, dissolved in a binary solvent mixture. Physicochemical and thermal characterization as well as biocompatibility analyses were carried out using SaOS-2 cells. The results showed fibers with highly porous surfaces whose pore diameters range in the nanometer scale and all scaffolds exhibited hydrophobicity. HA-modified scaffolds significantly improved cell metabolic activity and proliferation as compared to pristine scaffolds. The biodegradable and biocompatible scaffolds proposed in this study carry great potential for various biomedical applications and in the future, it is expected that they can be used for controlled drug delivery by incorporating growth factors, proteins, or drugs to reduce the inflammatory response and/or to promote bone repair.