Composite scaffolds, especially polymer/hydroxyapatite (HAP) composite scaffolds with predesigned structures, are promising materials for bone tissue engineering. Various methods including direct mixing of HAP powder with polymers or incubating polymer scaffolds in simulated body fluid for preparing polymer/HAP composite scaffolds are either uncontrolled or require long times of incubation. In this work, alginate/nano-HAP composite scaffolds with designed pore parameters and core/shell structures were fabricated using 3D plotting technique and in situ mineralization under mild conditions (at room temperature and without the use of any organic solvents). Light microscopy, scanning electron microscopy, microcomputer tomography, X-ray diffraction, and Fourier transform infrared spectroscopy were applied to characterize the fabricated scaffolds. Mechanical properties and protein delivery of the scaffolds were evaluated, as well as the cell response to the scaffolds by culturing human bone-marrow-derived mesenchymal stem cells (hBMSC). The obtained data indicate that this method is suitable to fabricate alginate/nano-HAP composite scaffolds with a layer of nano-HAP, coating the surface of the alginate strands homogeneously and completely. The surface mineralization enhanced the mechanical properties and improved the cell attachment and spreading, as well as supported sustaining protein release, compared to pure alginate scaffolds without nano-HAP shell layer. The results demonstrated that the method provides an interesting option for bone tissue engineering application.