Three-dimensional (3D) bioprinting technology enables the controlled deposition of cells and biomaterials (i.e., bioink) to easily create complex 3D biological microenvironments. Silk fibroin (SF) has recently emerged as a compelling bioink component due to its advantageous mechanical and biological properties. This study reports on the development and optimization of a novel bioink for extrusion-based 3D bioprinting and compares different bioink formulations based on mixtures of alginate methacrylate (ALMA), gelatin and SF. The rheological parameters of the bioink were investigated to predict printability and stability, and the optimal concentration of SF was selected. The bioink containing a low amount of SF (0.002% w/V) was found to be the best formulation. Light-assisted gelation of ALMA was exploited to obtain the final hydrogel matrix. Rheological analyses showed that SF-enriched hydrogels exhibited greater elasticity than SF-free hydrogels and were more tolerant to temperature fluctuations. Finally, MG-63 cells were successfully bioprinted and their viability and proliferation over time were analyzed. The SF-enriched bioink represents an excellent biomaterial in terms of printability and allows high cell proliferation over a period of up to 3 weeks. These data confirm the possibility of using the selected formulation for the successful bioprinting of cells into extracellular matrix-like microenvironments.