Three-dimensional microextrusion bioprinting has attracted great interest for fabrication of hierarchically structured, functional tissue substitutes with spatially defined cell distribution. Despite considerable progress, several significant limitations remain such as a lack of suitable bioinks which combine favorable cell response with high shape fidelity. Therefore, in this work a novel bioink of alginate-methylcellulose (AlgMC) blend functionalized with egg white (EW) was developed with the aim of solving this limitation. In this regard, a stepwise strategy was proposed to improve and examine the cell response in low-viscosity alginate inks (3%, w/v) with different EW concentrations, and in high-viscosity inks after gradual methylcellulose addition for enhancing printability. The rheological properties and printability of these cell-responsive bioinks were characterized to obtain an optimized formulation eliciting balanced physicochemical and biological properties for fabrication of volumetric scaffolds. The bioprinted AlgMC + EW constructs exhibited excellent shape fidelity while encapsulated human mesenchymal stem cells showed high post-printing viability as well as adhesion and spreading within the matrix. In a proof-of-concept experiment, the impact of these EW-mediated effects on osteogenesis of bioprinted primary human pre-osteoblasts (hOB) was evaluated. Results confirmed a high viability of hOB (93.7 ± 0.15%) post-fabrication in an EW-supported AlgMC bioink allowing cell adhesion, proliferation and migration. EW even promoted the expression of osteogenic genes, coding for bone sialoprotein (integrin binding sialoprotein/bone sialoprotein precursor (IBSP)) and osteocalcin (BGLAP) on mRNA level. To demonstrate the suitability of the novel ink for future fabrication of multi-zonal bone substitutes, AlgMC + EW was successfully co-printed together with a pasty calcium phosphate bone cement biomaterial ink to achieve a partly mineralized 3D volumetric environment with good cell viability and spreading. Along with the EW-mediated positive effects within bioprinted AlgMC-based scaffolds, this highlighted the promising potential of this novel ink for biofabrication of bone tissue substitutes in clinically relevant dimensions.